matrix for problem solving

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7 quick and easy steps to creating a decision matrix, with examples

Decisions, decisions, decisions. Making good decisions can help you steer your team in the right direction and hit your goals—but how do you know which decision is the right one? When faced with two seemingly equal choices, do you flip a coin? Roll the dice? Ask a Magic 8-Ball for help?

Decision-making is a critical part of good business planning—but it can be tricky to know which option is the right one. The key is making a quick decision without being hasty, and making the right decision without losing velocity.

What is a decision matrix?

A decision matrix is a tool to evaluate and select the best option between different choices. This tool is particularly useful if you are deciding between more than one option and there are several factors you need to consider in order to make your final decision.

You may have heard a decision matrix called by another term—even though they’re all talking about the same thing. Some other names for decision matrix include:

Pugh matrix

Grid analysis

Multi-attribute utility theory

Problem selection matrix

Decision grid

When to use a decision matrix

You don’t always need to use a decision matrix. This process is powerful—and relatively easy—but it’s most effective when you’re deciding between several comparable options. If the evaluation criteria aren’t the same between your different choices, then a decision matrix likely isn’t the best decision-making tool. For example, a decision matrix won’t help you decide what direction your team should take for the next year because the things you’re deciding between aren’t comparable.

Use a decision matrix if you are:

Comparing multiple, similar options

Narrowing down various options into one final decision

Weighing a variety of important factors

Hoping to approach the decision from a logical viewpoint, instead of an emotional or intuitive one

If a decision matrix isn’t right for your current situation, learn about other decision-making approaches below.

How to create a decision matrix in 7 steps

A decision matrix can help you evaluate the best option between different choices, based on several important factors and their relative importance. There are seven steps to creating a decision matrix:

1. Identify your alternatives

Decision matrices are a helpful tool to decide the best option between a set of similar choices. Before you can build your matrix, identify the options you’re deciding between.

For example, let’s say your team is launching a new brand campaign this summer. You need to decide on a vendor to create the visuals and videos for the design. Right now, you’ve identified three design agencies, though they each have their pros and cons.

2. Identify important considerations

The second step to building a decision matrix is to identify the important considerations that factor into your decision. This set of criteria helps you identify the best decision and avoid subjectivity.

Continuing our example, your team has decided that the important criteria to factor in when selecting a design agency are: cost, experience, communication, and past customer reviews.

3. Create your decision matrix

A decision matrix is a grid where you can compare important considerations between the various options.

Naturally, we build our decision matrices in Asana. Asana is a work management tool that can help you organize and execute work across your organization and provide the clarity teams need to hit their goals faster.

For example, here’s what your decision matrix skeleton looks like in Asana if you’re deciding between three agencies and factoring in cost, experience, communication, and customer reviews:

[Inline illustration] Empty decision matrix to decide between three design agencies (example)

4. Fill in your decision matrix

Now, rate each consideration on a predetermined scale. If there isn’t a large variation between the options, use a scale of 1-3, where three is the best. For more options, use a scale of 1-5, where five is the best.

This is where the advantages of a decision matrix really start to shine. For example, let’s say you’re deciding between three agencies and you have four important criteria, but you don’t make a decision matrix. Here’s how each agency stacks up:

Agency 1 is really cost effective but they don’t have a ton of experience. Their communication and customer reviews seem average.

Agency 2 isn’t very cost effective, but they aren’t the most expensive agency. They have a good amount of experience, and they have great customer reviews, but their communication so far has been a bit lacking.

Agency 3 is the most expensive, but they also have the most experience. Their communication so far has been average and their customer reviews are pretty good.

These three descriptions are all relatively similar—it’s hard to decide which is better based on a short paragraph, especially because each agency has its own pros and cons. Alternatively, here’s what the three agencies and their four considerations look like on a decision matrix when ranked from 1-5, with five being the best:

[Inline illustration] Decision matrix to decide between three design agencies with initial scores input (example)

5. Add weight

Sometimes, there are certain considerations that are more important than others. In such a case, use a weighted decision matrix to identify the best option for you.

To continue our example, imagine you absolutely can’t go over your budget, so cost is a critical factor in your decision-making process. Customer reviews are also important, since they give you a baseline sense of how effective each agency has been in the past.

To add weight to a decision matrix, assign a number (between 1-3 or 1-5, depending on how many options you have) to each consideration. Later in the decision-making process, you’ll multiply the weighting factor by each consideration.

Here’s what that looks like in our example:

[Inline illustration] Decision matrix to decide between three design agencies with weight (example)

6. Multiply the weighted score

Once you’ve applied your rating scale and assigned a weight to each consideration, multiply the weight by each consideration. This ensures that the more important considerations are being given more weight, which will ultimately help you select the best agency.

To continue our example, here’s what it looks like when you apply the weighted scores to each consideration for each agency:

[Inline illustration] Decision matrix to decide between three design agencies with weight multiplied by each number (example)

7. Calculate the total score

Now that you’ve multiplied the weighted score, add up all of the considerations for each agency. At this point, you should have a clear, numbers-based answer to which decision is the best one.

For example, this is what the finished decision matrix looks like:

[Inline illustration] Finished decision matrix to decide between three design agencies (example)

As you can see, Agency 2 has the highest score, so that is the agency you should go with. Even though Agency 1 was cheaper, the average cost of Agency 2, combined with their years of experience and stellar customer reviews make them the best option for your team. All that’s left is to contact the agency and move forward with the brand campaign.

Decision matrix example

You can use decision matrices for a variety of business decisions, as long as you’re weighing the best option between different choices. These decisions don’t always have to be business-critical, either. You can use this model to quickly make a simple decision as well.

For example, create a decision matrix to decide which chair you’re going to buy for your work from home setup. You like four different chairs, and your important considerations are comfort, cost, and reviews.

[Inline illustration] Finished decision matrix to decide between office chairs (example)

Decision-making alternatives

If the decision matrix method isn’t quite right for your choices, try:

Eisenhower matrix

An Eisenhower matrix is a 2x2 grid to help you prioritize tasks by urgency and importance. This matrix is helpful if you are juggling a variety of non-similar tasks and need to decide which tasks or initiatives to work on first.

In the upper left-hand corner, list urgent and important work: These tasks are a top priority. Do them now, or as soon as possible.

In the upper right-hand corner, list less urgent but important work: To ensure you get to these tasks, schedule them into your calendar, or capture the due date in a project management tool .

In the lower left-hand corner, list urgent and not important work: These tasks need to get done, but there is probably a better person for the job. Delegate this work if possible.

In the lower right-hand corner, list less urgent and not important work: Defer these tasks, or don’t do them. Clarifying your priorities and letting team members know that you can’t work on something right now is one way to reduce burnout .

Stakeholder analysis map and RACI chart

One of the most important decisions you have to make during the project planning process is to decide which stakeholders should be included, consulted, or informed. For this decision, create a stakeholder analysis map . This map helps you categorize stakeholders based on their relative influence and interest.

There are four categories in a stakeholder analysis map:

High influence and high interest: Involve these stakeholders in the project planning and decision-making process.

High influence and low interest: Let these stakeholders know about the project and monitor their interest in case they want to become more involved.

Low influence and high interest: Keep these stakeholders informed about the project. Add them to your project status updates so they can stay in the loop.

Low influence and low interest: Touch base with these stakeholders at regular checkpoints, but don’t worry too much about keeping them informed.

Once you’ve figured out your key stakeholders, you can also create a RACI chart . RACI is an acronym that stands for Responsible, Accountable, Consulted, and Informed. RACI charts can help you decide who the main decision-maker is for each task or initiative.

Team brainstorming session

Sometimes the best way to make a decision is to host a good old fashioned team brainstorm. Hold a whiteboard brainstorming session or share ideas in a project management tool.

At Asana, we like to use Kanban boards for dynamic brainstorming sessions. To start, the brainstorm facilitator creates a Board where team members can add ideas, thoughts, or feedback. Then, once everyone has added their ideas, each team member goes through and “likes” individual suggestions. Then, the team discusses the tasks with the most likes as a group to decide what to move forward on.

Say goodbye to coin flips for decision making

Making quick decisions is an important part of good project planning and project management . Whether you use a decision matrix to make a complex decision or a simple one, these tools can help you consider different factors and make the best decision for your team.

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35 problem-solving techniques and methods for solving complex problems

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How do you identify problems?

How do you identify the right solution.

Tips for more effective problem-solving

Complete problem-solving methods

Problem-solving techniques to identify and analyze problems
Problem-solving techniques for developing solutions

Problem-solving warm-up activities

Closing activities for a problem-solving process.

Before you can move towards finding the right solution for a given problem, you first need to identify and define the problem you wish to solve.

Here, you want to clearly articulate what the problem is and allow your group to do the same. Remember that everyone in a group is likely to have differing perspectives and alignment is necessary in order to help the group move forward.

Identifying a problem accurately also requires that all members of a group are able to contribute their views in an open and safe manner. It can be scary for people to stand up and contribute, especially if the problems or challenges are emotive or personal in nature. Be sure to try and create a psychologically safe space for these kinds of discussions.

Remember that problem analysis and further discussion are also important. Not taking the time to fully analyze and discuss a challenge can result in the development of solutions that are not fit for purpose or do not address the underlying issue.

Successfully identifying and then analyzing a problem means facilitating a group through activities designed to help them clearly and honestly articulate their thoughts and produce usable insight.

With this data, you might then produce a problem statement that clearly describes the problem you wish to be addressed and also state the goal of any process you undertake to tackle this issue.

Finding solutions is the end goal of any process. Complex organizational challenges can only be solved with an appropriate solution but discovering them requires using the right problem-solving tool.

After you’ve explored a problem and discussed ideas, you need to help a team discuss and choose the right solution. Consensus tools and methods such as those below help a group explore possible solutions before then voting for the best. They’re a great way to tap into the collective intelligence of the group for great results!

Remember that the process is often iterative. Great problem solvers often roadtest a viable solution in a measured way to see what works too. While you might not get the right solution on your first try, the methods below help teams land on the most likely to succeed solution while also holding space for improvement.

Every effective problem solving process begins with an agenda . A well-structured workshop is one of the best methods for successfully guiding a group from exploring a problem to implementing a solution.

In SessionLab, it’s easy to go from an idea to a complete agenda . Start by dragging and dropping your core problem solving activities into place . Add timings, breaks and necessary materials before sharing your agenda with your colleagues.

The resulting agenda will be your guide to an effective and productive problem solving session that will also help you stay organized on the day!

Tips for more effective problem solving

Problem-solving activities are only one part of the puzzle. While a great method can help unlock your team’s ability to solve problems, without a thoughtful approach and strong facilitation the solutions may not be fit for purpose.

Let’s take a look at some problem-solving tips you can apply to any process to help it be a success!

Clearly define the problem

Jumping straight to solutions can be tempting, though without first clearly articulating a problem, the solution might not be the right one. Many of the problem-solving activities below include sections where the problem is explored and clearly defined before moving on.

This is a vital part of the problem-solving process and taking the time to fully define an issue can save time and effort later. A clear definition helps identify irrelevant information and it also ensures that your team sets off on the right track.

Don’t jump to conclusions

It’s easy for groups to exhibit cognitive bias or have preconceived ideas about both problems and potential solutions. Be sure to back up any problem statements or potential solutions with facts, research, and adequate forethought.

The best techniques ask participants to be methodical and challenge preconceived notions. Make sure you give the group enough time and space to collect relevant information and consider the problem in a new way. By approaching the process with a clear, rational mindset, you’ll often find that better solutions are more forthcoming.

Try different approaches

Problems come in all shapes and sizes and so too should the methods you use to solve them. If you find that one approach isn’t yielding results and your team isn’t finding different solutions, try mixing it up. You’ll be surprised at how using a new creative activity can unblock your team and generate great solutions.

Don’t take it personally

Depending on the nature of your team or organizational problems, it’s easy for conversations to get heated. While it’s good for participants to be engaged in the discussions, ensure that emotions don’t run too high and that blame isn’t thrown around while finding solutions.

You’re all in it together, and even if your team or area is seeing problems, that isn’t necessarily a disparagement of you personally. Using facilitation skills to manage group dynamics is one effective method of helping conversations be more constructive.

Get the right people in the room

Your problem-solving method is often only as effective as the group using it. Getting the right people on the job and managing the number of people present is important too!

If the group is too small, you may not get enough different perspectives to effectively solve a problem. If the group is too large, you can go round and round during the ideation stages.

Creating the right group makeup is also important in ensuring you have the necessary expertise and skillset to both identify and follow up on potential solutions. Carefully consider who to include at each stage to help ensure your problem-solving method is followed and positioned for success.

Document everything

The best solutions can take refinement, iteration, and reflection to come out. Get into a habit of documenting your process in order to keep all the learnings from the session and to allow ideas to mature and develop. Many of the methods below involve the creation of documents or shared resources. Be sure to keep and share these so everyone can benefit from the work done!

Bring a facilitator

Facilitation is all about making group processes easier. With a subject as potentially emotive and important as problem-solving, having an impartial third party in the form of a facilitator can make all the difference in finding great solutions and keeping the process moving. Consider bringing a facilitator to your problem-solving session to get better results and generate meaningful solutions!

Develop your problem-solving skills

It takes time and practice to be an effective problem solver. While some roles or participants might more naturally gravitate towards problem-solving, it can take development and planning to help everyone create better solutions.

You might develop a training program, run a problem-solving workshop or simply ask your team to practice using the techniques below. Check out our post on problem-solving skills to see how you and your group can develop the right mental process and be more resilient to issues too!

Design a great agenda

Workshops are a great format for solving problems. With the right approach, you can focus a group and help them find the solutions to their own problems. But designing a process can be time-consuming and finding the right activities can be difficult.

Check out our workshop planning guide to level-up your agenda design and start running more effective workshops. Need inspiration? Check out templates designed by expert facilitators to help you kickstart your process!

In this section, we’ll look at in-depth problem-solving methods that provide a complete end-to-end process for developing effective solutions. These will help guide your team from the discovery and definition of a problem through to delivering the right solution.

If you’re looking for an all-encompassing method or problem-solving model, these processes are a great place to start. They’ll ask your team to challenge preconceived ideas and adopt a mindset for solving problems more effectively.

Six Thinking Hats
Lightning Decision Jam
Problem Definition Process
Discovery & Action Dialogue

Design Sprint 2.0

Open Space Technology

1. Six Thinking Hats

Individual approaches to solving a problem can be very different based on what team or role an individual holds. It can be easy for existing biases or perspectives to find their way into the mix, or for internal politics to direct a conversation.

Six Thinking Hats is a classic method for identifying the problems that need to be solved and enables your team to consider them from different angles, whether that is by focusing on facts and data, creative solutions, or by considering why a particular solution might not work.

Like all problem-solving frameworks, Six Thinking Hats is effective at helping teams remove roadblocks from a conversation or discussion and come to terms with all the aspects necessary to solve complex problems.

2. Lightning Decision Jam

Featured courtesy of Jonathan Courtney of AJ&Smart Berlin, Lightning Decision Jam is one of those strategies that should be in every facilitation toolbox. Exploring problems and finding solutions is often creative in nature, though as with any creative process, there is the potential to lose focus and get lost.

Unstructured discussions might get you there in the end, but it’s much more effective to use a method that creates a clear process and team focus.

In Lightning Decision Jam, participants are invited to begin by writing challenges, concerns, or mistakes on post-its without discussing them before then being invited by the moderator to present them to the group.

From there, the team vote on which problems to solve and are guided through steps that will allow them to reframe those problems, create solutions and then decide what to execute on.

By deciding the problems that need to be solved as a team before moving on, this group process is great for ensuring the whole team is aligned and can take ownership over the next stages.

Lightning Decision Jam (LDJ) #action #decision making #problem solving #issue analysis #innovation #design #remote-friendly The problem with anything that requires creative thinking is that it’s easy to get lost—lose focus and fall into the trap of having useless, open-ended, unstructured discussions. Here’s the most effective solution I’ve found: Replace all open, unstructured discussion with a clear process. What to use this exercise for: Anything which requires a group of people to make decisions, solve problems or discuss challenges. It’s always good to frame an LDJ session with a broad topic, here are some examples: The conversion flow of our checkout Our internal design process How we organise events Keeping up with our competition Improving sales flow

3. Problem Definition Process

While problems can be complex, the problem-solving methods you use to identify and solve those problems can often be simple in design.

By taking the time to truly identify and define a problem before asking the group to reframe the challenge as an opportunity, this method is a great way to enable change.

Begin by identifying a focus question and exploring the ways in which it manifests before splitting into five teams who will each consider the problem using a different method: escape, reversal, exaggeration, distortion or wishful. Teams develop a problem objective and create ideas in line with their method before then feeding them back to the group.

This method is great for enabling in-depth discussions while also creating space for finding creative solutions too!

Problem Definition #problem solving #idea generation #creativity #online #remote-friendly A problem solving technique to define a problem, challenge or opportunity and to generate ideas.

4. The 5 Whys

Sometimes, a group needs to go further with their strategies and analyze the root cause at the heart of organizational issues. An RCA or root cause analysis is the process of identifying what is at the heart of business problems or recurring challenges.

The 5 Whys is a simple and effective method of helping a group go find the root cause of any problem or challenge and conduct analysis that will deliver results.

By beginning with the creation of a problem statement and going through five stages to refine it, The 5 Whys provides everything you need to truly discover the cause of an issue.

The 5 Whys #hyperisland #innovation This simple and powerful method is useful for getting to the core of a problem or challenge. As the title suggests, the group defines a problems, then asks the question “why” five times, often using the resulting explanation as a starting point for creative problem solving.

5. World Cafe

World Cafe is a simple but powerful facilitation technique to help bigger groups to focus their energy and attention on solving complex problems.

World Cafe enables this approach by creating a relaxed atmosphere where participants are able to self-organize and explore topics relevant and important to them which are themed around a central problem-solving purpose. Create the right atmosphere by modeling your space after a cafe and after guiding the group through the method, let them take the lead!

Making problem-solving a part of your organization’s culture in the long term can be a difficult undertaking. More approachable formats like World Cafe can be especially effective in bringing people unfamiliar with workshops into the fold.

World Cafe #hyperisland #innovation #issue analysis World Café is a simple yet powerful method, originated by Juanita Brown, for enabling meaningful conversations driven completely by participants and the topics that are relevant and important to them. Facilitators create a cafe-style space and provide simple guidelines. Participants then self-organize and explore a set of relevant topics or questions for conversation.

6. Discovery & Action Dialogue (DAD)

One of the best approaches is to create a safe space for a group to share and discover practices and behaviors that can help them find their own solutions.

With DAD, you can help a group choose which problems they wish to solve and which approaches they will take to do so. It’s great at helping remove resistance to change and can help get buy-in at every level too!

This process of enabling frontline ownership is great in ensuring follow-through and is one of the methods you will want in your toolbox as a facilitator.

Discovery & Action Dialogue (DAD) #idea generation #liberating structures #action #issue analysis #remote-friendly DADs make it easy for a group or community to discover practices and behaviors that enable some individuals (without access to special resources and facing the same constraints) to find better solutions than their peers to common problems. These are called positive deviant (PD) behaviors and practices. DADs make it possible for people in the group, unit, or community to discover by themselves these PD practices. DADs also create favorable conditions for stimulating participants’ creativity in spaces where they can feel safe to invent new and more effective practices. Resistance to change evaporates as participants are unleashed to choose freely which practices they will adopt or try and which problems they will tackle. DADs make it possible to achieve frontline ownership of solutions.

7. Design Sprint 2.0

Want to see how a team can solve big problems and move forward with prototyping and testing solutions in a few days? The Design Sprint 2.0 template from Jake Knapp, author of Sprint, is a complete agenda for a with proven results.

Developing the right agenda can involve difficult but necessary planning. Ensuring all the correct steps are followed can also be stressful or time-consuming depending on your level of experience.

Use this complete 4-day workshop template if you are finding there is no obvious solution to your challenge and want to focus your team around a specific problem that might require a shortcut to launching a minimum viable product or waiting for the organization-wide implementation of a solution.

8. Open space technology

Open space technology- developed by Harrison Owen – creates a space where large groups are invited to take ownership of their problem solving and lead individual sessions. Open space technology is a great format when you have a great deal of expertise and insight in the room and want to allow for different takes and approaches on a particular theme or problem you need to be solved.

Start by bringing your participants together to align around a central theme and focus their efforts. Explain the ground rules to help guide the problem-solving process and then invite members to identify any issue connecting to the central theme that they are interested in and are prepared to take responsibility for.

Once participants have decided on their approach to the core theme, they write their issue on a piece of paper, announce it to the group, pick a session time and place, and post the paper on the wall. As the wall fills up with sessions, the group is then invited to join the sessions that interest them the most and which they can contribute to, then you’re ready to begin!

Everyone joins the problem-solving group they’ve signed up to, record the discussion and if appropriate, findings can then be shared with the rest of the group afterward.

Open Space Technology #action plan #idea generation #problem solving #issue analysis #large group #online #remote-friendly Open Space is a methodology for large groups to create their agenda discerning important topics for discussion, suitable for conferences, community gatherings and whole system facilitation

Techniques to identify and analyze problems

Using a problem-solving method to help a team identify and analyze a problem can be a quick and effective addition to any workshop or meeting.

While further actions are always necessary, you can generate momentum and alignment easily, and these activities are a great place to get started.

We’ve put together this list of techniques to help you and your team with problem identification, analysis, and discussion that sets the foundation for developing effective solutions.

Let’s take a look!

The Creativity Dice
Fishbone Analysis
Problem Tree
SWOT Analysis
Agreement-Certainty Matrix
The Journalistic Six
LEGO Challenge
What, So What, Now What?
Journalists

Individual and group perspectives are incredibly important, but what happens if people are set in their minds and need a change of perspective in order to approach a problem more effectively?

Flip It is a method we love because it is both simple to understand and run, and allows groups to understand how their perspectives and biases are formed.

Participants in Flip It are first invited to consider concerns, issues, or problems from a perspective of fear and write them on a flip chart. Then, the group is asked to consider those same issues from a perspective of hope and flip their understanding.

No problem and solution is free from existing bias and by changing perspectives with Flip It, you can then develop a problem solving model quickly and effectively.

Flip It! #gamestorming #problem solving #action Often, a change in a problem or situation comes simply from a change in our perspectives. Flip It! is a quick game designed to show players that perspectives are made, not born.

10. The Creativity Dice

One of the most useful problem solving skills you can teach your team is of approaching challenges with creativity, flexibility, and openness. Games like The Creativity Dice allow teams to overcome the potential hurdle of too much linear thinking and approach the process with a sense of fun and speed.

In The Creativity Dice, participants are organized around a topic and roll a dice to determine what they will work on for a period of 3 minutes at a time. They might roll a 3 and work on investigating factual information on the chosen topic. They might roll a 1 and work on identifying the specific goals, standards, or criteria for the session.

Encouraging rapid work and iteration while asking participants to be flexible are great skills to cultivate. Having a stage for idea incubation in this game is also important. Moments of pause can help ensure the ideas that are put forward are the most suitable.

The Creativity Dice #creativity #problem solving #thiagi #issue analysis Too much linear thinking is hazardous to creative problem solving. To be creative, you should approach the problem (or the opportunity) from different points of view. You should leave a thought hanging in mid-air and move to another. This skipping around prevents premature closure and lets your brain incubate one line of thought while you consciously pursue another.

11. Fishbone Analysis

Organizational or team challenges are rarely simple, and it’s important to remember that one problem can be an indication of something that goes deeper and may require further consideration to be solved.

Fishbone Analysis helps groups to dig deeper and understand the origins of a problem. It’s a great example of a root cause analysis method that is simple for everyone on a team to get their head around.

Participants in this activity are asked to annotate a diagram of a fish, first adding the problem or issue to be worked on at the head of a fish before then brainstorming the root causes of the problem and adding them as bones on the fish.

Using abstractions such as a diagram of a fish can really help a team break out of their regular thinking and develop a creative approach.

Fishbone Analysis #problem solving ##root cause analysis #decision making #online facilitation A process to help identify and understand the origins of problems, issues or observations.

12. Problem Tree

Encouraging visual thinking can be an essential part of many strategies. By simply reframing and clarifying problems, a group can move towards developing a problem solving model that works for them.

In Problem Tree, groups are asked to first brainstorm a list of problems – these can be design problems, team problems or larger business problems – and then organize them into a hierarchy. The hierarchy could be from most important to least important or abstract to practical, though the key thing with problem solving games that involve this aspect is that your group has some way of managing and sorting all the issues that are raised.

Once you have a list of problems that need to be solved and have organized them accordingly, you’re then well-positioned for the next problem solving steps.

Problem tree #define intentions #create #design #issue analysis A problem tree is a tool to clarify the hierarchy of problems addressed by the team within a design project; it represents high level problems or related sublevel problems.

13. SWOT Analysis

Chances are you’ve heard of the SWOT Analysis before. This problem-solving method focuses on identifying strengths, weaknesses, opportunities, and threats is a tried and tested method for both individuals and teams.

Start by creating a desired end state or outcome and bare this in mind – any process solving model is made more effective by knowing what you are moving towards. Create a quadrant made up of the four categories of a SWOT analysis and ask participants to generate ideas based on each of those quadrants.

Once you have those ideas assembled in their quadrants, cluster them together based on their affinity with other ideas. These clusters are then used to facilitate group conversations and move things forward.

SWOT analysis #gamestorming #problem solving #action #meeting facilitation The SWOT Analysis is a long-standing technique of looking at what we have, with respect to the desired end state, as well as what we could improve on. It gives us an opportunity to gauge approaching opportunities and dangers, and assess the seriousness of the conditions that affect our future. When we understand those conditions, we can influence what comes next.

14. Agreement-Certainty Matrix

Not every problem-solving approach is right for every challenge, and deciding on the right method for the challenge at hand is a key part of being an effective team.

The Agreement Certainty matrix helps teams align on the nature of the challenges facing them. By sorting problems from simple to chaotic, your team can understand what methods are suitable for each problem and what they can do to ensure effective results.

If you are already using Liberating Structures techniques as part of your problem-solving strategy, the Agreement-Certainty Matrix can be an invaluable addition to your process. We’ve found it particularly if you are having issues with recurring problems in your organization and want to go deeper in understanding the root cause.

Agreement-Certainty Matrix #issue analysis #liberating structures #problem solving You can help individuals or groups avoid the frequent mistake of trying to solve a problem with methods that are not adapted to the nature of their challenge. The combination of two questions makes it possible to easily sort challenges into four categories: simple, complicated, complex , and chaotic . A problem is simple when it can be solved reliably with practices that are easy to duplicate. It is complicated when experts are required to devise a sophisticated solution that will yield the desired results predictably. A problem is complex when there are several valid ways to proceed but outcomes are not predictable in detail. Chaotic is when the context is too turbulent to identify a path forward. A loose analogy may be used to describe these differences: simple is like following a recipe, complicated like sending a rocket to the moon, complex like raising a child, and chaotic is like the game “Pin the Tail on the Donkey.” The Liberating Structures Matching Matrix in Chapter 5 can be used as the first step to clarify the nature of a challenge and avoid the mismatches between problems and solutions that are frequently at the root of chronic, recurring problems.

Organizing and charting a team’s progress can be important in ensuring its success. SQUID (Sequential Question and Insight Diagram) is a great model that allows a team to effectively switch between giving questions and answers and develop the skills they need to stay on track throughout the process.

Begin with two different colored sticky notes – one for questions and one for answers – and with your central topic (the head of the squid) on the board. Ask the group to first come up with a series of questions connected to their best guess of how to approach the topic. Ask the group to come up with answers to those questions, fix them to the board and connect them with a line. After some discussion, go back to question mode by responding to the generated answers or other points on the board.

It’s rewarding to see a diagram grow throughout the exercise, and a completed SQUID can provide a visual resource for future effort and as an example for other teams.

SQUID #gamestorming #project planning #issue analysis #problem solving When exploring an information space, it’s important for a group to know where they are at any given time. By using SQUID, a group charts out the territory as they go and can navigate accordingly. SQUID stands for Sequential Question and Insight Diagram.

16. Speed Boat

To continue with our nautical theme, Speed Boat is a short and sweet activity that can help a team quickly identify what employees, clients or service users might have a problem with and analyze what might be standing in the way of achieving a solution.

Methods that allow for a group to make observations, have insights and obtain those eureka moments quickly are invaluable when trying to solve complex problems.

In Speed Boat, the approach is to first consider what anchors and challenges might be holding an organization (or boat) back. Bonus points if you are able to identify any sharks in the water and develop ideas that can also deal with competitors!

Speed Boat #gamestorming #problem solving #action Speedboat is a short and sweet way to identify what your employees or clients don’t like about your product/service or what’s standing in the way of a desired goal.

17. The Journalistic Six

Some of the most effective ways of solving problems is by encouraging teams to be more inclusive and diverse in their thinking.

Based on the six key questions journalism students are taught to answer in articles and news stories, The Journalistic Six helps create teams to see the whole picture. By using who, what, when, where, why, and how to facilitate the conversation and encourage creative thinking, your team can make sure that the problem identification and problem analysis stages of the are covered exhaustively and thoughtfully. Reporter’s notebook and dictaphone optional.

The Journalistic Six – Who What When Where Why How #idea generation #issue analysis #problem solving #online #creative thinking #remote-friendly A questioning method for generating, explaining, investigating ideas.

18. LEGO Challenge

Now for an activity that is a little out of the (toy) box. LEGO Serious Play is a facilitation methodology that can be used to improve creative thinking and problem-solving skills.

The LEGO Challenge includes giving each member of the team an assignment that is hidden from the rest of the group while they create a structure without speaking.

What the LEGO challenge brings to the table is a fun working example of working with stakeholders who might not be on the same page to solve problems. Also, it’s LEGO! Who doesn’t love LEGO!

LEGO Challenge #hyperisland #team A team-building activity in which groups must work together to build a structure out of LEGO, but each individual has a secret “assignment” which makes the collaborative process more challenging. It emphasizes group communication, leadership dynamics, conflict, cooperation, patience and problem solving strategy.

19. What, So What, Now What?

If not carefully managed, the problem identification and problem analysis stages of the problem-solving process can actually create more problems and misunderstandings.

The What, So What, Now What? problem-solving activity is designed to help collect insights and move forward while also eliminating the possibility of disagreement when it comes to identifying, clarifying, and analyzing organizational or work problems.

Facilitation is all about bringing groups together so that might work on a shared goal and the best problem-solving strategies ensure that teams are aligned in purpose, if not initially in opinion or insight.

Throughout the three steps of this game, you give everyone on a team to reflect on a problem by asking what happened, why it is important, and what actions should then be taken.

This can be a great activity for bringing our individual perceptions about a problem or challenge and contextualizing it in a larger group setting. This is one of the most important problem-solving skills you can bring to your organization.

W³ – What, So What, Now What? #issue analysis #innovation #liberating structures You can help groups reflect on a shared experience in a way that builds understanding and spurs coordinated action while avoiding unproductive conflict. It is possible for every voice to be heard while simultaneously sifting for insights and shaping new direction. Progressing in stages makes this practical—from collecting facts about What Happened to making sense of these facts with So What and finally to what actions logically follow with Now What . The shared progression eliminates most of the misunderstandings that otherwise fuel disagreements about what to do. Voila!

20. Journalists

Problem analysis can be one of the most important and decisive stages of all problem-solving tools. Sometimes, a team can become bogged down in the details and are unable to move forward.

Journalists is an activity that can avoid a group from getting stuck in the problem identification or problem analysis stages of the process.

In Journalists, the group is invited to draft the front page of a fictional newspaper and figure out what stories deserve to be on the cover and what headlines those stories will have. By reframing how your problems and challenges are approached, you can help a team move productively through the process and be better prepared for the steps to follow.

Journalists #vision #big picture #issue analysis #remote-friendly This is an exercise to use when the group gets stuck in details and struggles to see the big picture. Also good for defining a vision.

Problem-solving techniques for developing solutions

The success of any problem-solving process can be measured by the solutions it produces. After you’ve defined the issue, explored existing ideas, and ideated, it’s time to narrow down to the correct solution.

Use these problem-solving techniques when you want to help your team find consensus, compare possible solutions, and move towards taking action on a particular problem.

Improved Solutions
Four-Step Sketch
15% Solutions
How-Now-Wow matrix
Impact Effort Matrix

21. Mindspin

Brainstorming is part of the bread and butter of the problem-solving process and all problem-solving strategies benefit from getting ideas out and challenging a team to generate solutions quickly.

With Mindspin, participants are encouraged not only to generate ideas but to do so under time constraints and by slamming down cards and passing them on. By doing multiple rounds, your team can begin with a free generation of possible solutions before moving on to developing those solutions and encouraging further ideation.

This is one of our favorite problem-solving activities and can be great for keeping the energy up throughout the workshop. Remember the importance of helping people become engaged in the process – energizing problem-solving techniques like Mindspin can help ensure your team stays engaged and happy, even when the problems they’re coming together to solve are complex.

MindSpin #teampedia #idea generation #problem solving #action A fast and loud method to enhance brainstorming within a team. Since this activity has more than round ideas that are repetitive can be ruled out leaving more creative and innovative answers to the challenge.

22. Improved Solutions

After a team has successfully identified a problem and come up with a few solutions, it can be tempting to call the work of the problem-solving process complete. That said, the first solution is not necessarily the best, and by including a further review and reflection activity into your problem-solving model, you can ensure your group reaches the best possible result.

One of a number of problem-solving games from Thiagi Group, Improved Solutions helps you go the extra mile and develop suggested solutions with close consideration and peer review. By supporting the discussion of several problems at once and by shifting team roles throughout, this problem-solving technique is a dynamic way of finding the best solution.

Improved Solutions #creativity #thiagi #problem solving #action #team You can improve any solution by objectively reviewing its strengths and weaknesses and making suitable adjustments. In this creativity framegame, you improve the solutions to several problems. To maintain objective detachment, you deal with a different problem during each of six rounds and assume different roles (problem owner, consultant, basher, booster, enhancer, and evaluator) during each round. At the conclusion of the activity, each player ends up with two solutions to her problem.

23. Four Step Sketch

Creative thinking and visual ideation does not need to be confined to the opening stages of your problem-solving strategies. Exercises that include sketching and prototyping on paper can be effective at the solution finding and development stage of the process, and can be great for keeping a team engaged.

By going from simple notes to a crazy 8s round that involves rapidly sketching 8 variations on their ideas before then producing a final solution sketch, the group is able to iterate quickly and visually. Problem-solving techniques like Four-Step Sketch are great if you have a group of different thinkers and want to change things up from a more textual or discussion-based approach.

Four-Step Sketch #design sprint #innovation #idea generation #remote-friendly The four-step sketch is an exercise that helps people to create well-formed concepts through a structured process that includes: Review key information Start design work on paper, Consider multiple variations , Create a detailed solution . This exercise is preceded by a set of other activities allowing the group to clarify the challenge they want to solve. See how the Four Step Sketch exercise fits into a Design Sprint

24. 15% Solutions

Some problems are simpler than others and with the right problem-solving activities, you can empower people to take immediate actions that can help create organizational change.

Part of the liberating structures toolkit, 15% solutions is a problem-solving technique that focuses on finding and implementing solutions quickly. A process of iterating and making small changes quickly can help generate momentum and an appetite for solving complex problems.

Problem-solving strategies can live and die on whether people are onboard. Getting some quick wins is a great way of getting people behind the process.

It can be extremely empowering for a team to realize that problem-solving techniques can be deployed quickly and easily and delineate between things they can positively impact and those things they cannot change.

15% Solutions #action #liberating structures #remote-friendly You can reveal the actions, however small, that everyone can do immediately. At a minimum, these will create momentum, and that may make a BIG difference. 15% Solutions show that there is no reason to wait around, feel powerless, or fearful. They help people pick it up a level. They get individuals and the group to focus on what is within their discretion instead of what they cannot change. With a very simple question, you can flip the conversation to what can be done and find solutions to big problems that are often distributed widely in places not known in advance. Shifting a few grains of sand may trigger a landslide and change the whole landscape.

25. How-Now-Wow Matrix

The problem-solving process is often creative, as complex problems usually require a change of thinking and creative response in order to find the best solutions. While it’s common for the first stages to encourage creative thinking, groups can often gravitate to familiar solutions when it comes to the end of the process.

When selecting solutions, you don’t want to lose your creative energy! The How-Now-Wow Matrix from Gamestorming is a great problem-solving activity that enables a group to stay creative and think out of the box when it comes to selecting the right solution for a given problem.

Problem-solving techniques that encourage creative thinking and the ideation and selection of new solutions can be the most effective in organisational change. Give the How-Now-Wow Matrix a go, and not just for how pleasant it is to say out loud.

How-Now-Wow Matrix #gamestorming #idea generation #remote-friendly When people want to develop new ideas, they most often think out of the box in the brainstorming or divergent phase. However, when it comes to convergence, people often end up picking ideas that are most familiar to them. This is called a ‘creative paradox’ or a ‘creadox’. The How-Now-Wow matrix is an idea selection tool that breaks the creadox by forcing people to weigh each idea on 2 parameters.

26. Impact and Effort Matrix

All problem-solving techniques hope to not only find solutions to a given problem or challenge but to find the best solution. When it comes to finding a solution, groups are invited to put on their decision-making hats and really think about how a proposed idea would work in practice.

The Impact and Effort Matrix is one of the problem-solving techniques that fall into this camp, empowering participants to first generate ideas and then categorize them into a 2×2 matrix based on impact and effort.

Activities that invite critical thinking while remaining simple are invaluable. Use the Impact and Effort Matrix to move from ideation and towards evaluating potential solutions before then committing to them.

Impact and Effort Matrix #gamestorming #decision making #action #remote-friendly In this decision-making exercise, possible actions are mapped based on two factors: effort required to implement and potential impact. Categorizing ideas along these lines is a useful technique in decision making, as it obliges contributors to balance and evaluate suggested actions before committing to them.

27. Dotmocracy

If you’ve followed each of the problem-solving steps with your group successfully, you should move towards the end of your process with heaps of possible solutions developed with a specific problem in mind. But how do you help a group go from ideation to putting a solution into action?

Dotmocracy – or Dot Voting -is a tried and tested method of helping a team in the problem-solving process make decisions and put actions in place with a degree of oversight and consensus.

One of the problem-solving techniques that should be in every facilitator’s toolbox, Dot Voting is fast and effective and can help identify the most popular and best solutions and help bring a group to a decision effectively.

Dotmocracy #action #decision making #group prioritization #hyperisland #remote-friendly Dotmocracy is a simple method for group prioritization or decision-making. It is not an activity on its own, but a method to use in processes where prioritization or decision-making is the aim. The method supports a group to quickly see which options are most popular or relevant. The options or ideas are written on post-its and stuck up on a wall for the whole group to see. Each person votes for the options they think are the strongest, and that information is used to inform a decision.

All facilitators know that warm-ups and icebreakers are useful for any workshop or group process. Problem-solving workshops are no different.

Use these problem-solving techniques to warm up a group and prepare them for the rest of the process. Activating your group by tapping into some of the top problem-solving skills can be one of the best ways to see great outcomes from your session.

Check-in/Check-out
Doodling Together
Show and Tell
Constellations
Draw a Tree

28. Check-in / Check-out

Solid processes are planned from beginning to end, and the best facilitators know that setting the tone and establishing a safe, open environment can be integral to a successful problem-solving process.

Check-in / Check-out is a great way to begin and/or bookend a problem-solving workshop. Checking in to a session emphasizes that everyone will be seen, heard, and expected to contribute.

If you are running a series of meetings, setting a consistent pattern of checking in and checking out can really help your team get into a groove. We recommend this opening-closing activity for small to medium-sized groups though it can work with large groups if they’re disciplined!

Check-in / Check-out #team #opening #closing #hyperisland #remote-friendly Either checking-in or checking-out is a simple way for a team to open or close a process, symbolically and in a collaborative way. Checking-in/out invites each member in a group to be present, seen and heard, and to express a reflection or a feeling. Checking-in emphasizes presence, focus and group commitment; checking-out emphasizes reflection and symbolic closure.

29. Doodling Together

Thinking creatively and not being afraid to make suggestions are important problem-solving skills for any group or team, and warming up by encouraging these behaviors is a great way to start.

Doodling Together is one of our favorite creative ice breaker games – it’s quick, effective, and fun and can make all following problem-solving steps easier by encouraging a group to collaborate visually. By passing cards and adding additional items as they go, the workshop group gets into a groove of co-creation and idea development that is crucial to finding solutions to problems.

Doodling Together #collaboration #creativity #teamwork #fun #team #visual methods #energiser #icebreaker #remote-friendly Create wild, weird and often funny postcards together & establish a group’s creative confidence.

30. Show and Tell

You might remember some version of Show and Tell from being a kid in school and it’s a great problem-solving activity to kick off a session.

Asking participants to prepare a little something before a workshop by bringing an object for show and tell can help them warm up before the session has even begun! Games that include a physical object can also help encourage early engagement before moving onto more big-picture thinking.

By asking your participants to tell stories about why they chose to bring a particular item to the group, you can help teams see things from new perspectives and see both differences and similarities in the way they approach a topic. Great groundwork for approaching a problem-solving process as a team!

Show and Tell #gamestorming #action #opening #meeting facilitation Show and Tell taps into the power of metaphors to reveal players’ underlying assumptions and associations around a topic The aim of the game is to get a deeper understanding of stakeholders’ perspectives on anything—a new project, an organizational restructuring, a shift in the company’s vision or team dynamic.

31. Constellations

Who doesn’t love stars? Constellations is a great warm-up activity for any workshop as it gets people up off their feet, energized, and ready to engage in new ways with established topics. It’s also great for showing existing beliefs, biases, and patterns that can come into play as part of your session.

Using warm-up games that help build trust and connection while also allowing for non-verbal responses can be great for easing people into the problem-solving process and encouraging engagement from everyone in the group. Constellations is great in large spaces that allow for movement and is definitely a practical exercise to allow the group to see patterns that are otherwise invisible.

Constellations #trust #connection #opening #coaching #patterns #system Individuals express their response to a statement or idea by standing closer or further from a central object. Used with teams to reveal system, hidden patterns, perspectives.

32. Draw a Tree

Problem-solving games that help raise group awareness through a central, unifying metaphor can be effective ways to warm-up a group in any problem-solving model.

Draw a Tree is a simple warm-up activity you can use in any group and which can provide a quick jolt of energy. Start by asking your participants to draw a tree in just 45 seconds – they can choose whether it will be abstract or realistic.

Once the timer is up, ask the group how many people included the roots of the tree and use this as a means to discuss how we can ignore important parts of any system simply because they are not visible.

All problem-solving strategies are made more effective by thinking of problems critically and by exposing things that may not normally come to light. Warm-up games like Draw a Tree are great in that they quickly demonstrate some key problem-solving skills in an accessible and effective way.

Draw a Tree #thiagi #opening #perspectives #remote-friendly With this game you can raise awarness about being more mindful, and aware of the environment we live in.

Each step of the problem-solving workshop benefits from an intelligent deployment of activities, games, and techniques. Bringing your session to an effective close helps ensure that solutions are followed through on and that you also celebrate what has been achieved.

Here are some problem-solving activities you can use to effectively close a workshop or meeting and ensure the great work you’ve done can continue afterward.

One Breath Feedback
Who What When Matrix
Response Cards

How do I conclude a problem-solving process?

All good things must come to an end. With the bulk of the work done, it can be tempting to conclude your workshop swiftly and without a moment to debrief and align. This can be problematic in that it doesn’t allow your team to fully process the results or reflect on the process.

At the end of an effective session, your team will have gone through a process that, while productive, can be exhausting. It’s important to give your group a moment to take a breath, ensure that they are clear on future actions, and provide short feedback before leaving the space.

The primary purpose of any problem-solving method is to generate solutions and then implement them. Be sure to take the opportunity to ensure everyone is aligned and ready to effectively implement the solutions you produced in the workshop.

Remember that every process can be improved and by giving a short moment to collect feedback in the session, you can further refine your problem-solving methods and see further success in the future too.

33. One Breath Feedback

Maintaining attention and focus during the closing stages of a problem-solving workshop can be tricky and so being concise when giving feedback can be important. It’s easy to incur “death by feedback” should some team members go on for too long sharing their perspectives in a quick feedback round.

One Breath Feedback is a great closing activity for workshops. You give everyone an opportunity to provide feedback on what they’ve done but only in the space of a single breath. This keeps feedback short and to the point and means that everyone is encouraged to provide the most important piece of feedback to them.

One breath feedback #closing #feedback #action This is a feedback round in just one breath that excels in maintaining attention: each participants is able to speak during just one breath … for most people that’s around 20 to 25 seconds … unless of course you’ve been a deep sea diver in which case you’ll be able to do it for longer.

34. Who What When Matrix

Matrices feature as part of many effective problem-solving strategies and with good reason. They are easily recognizable, simple to use, and generate results.

The Who What When Matrix is a great tool to use when closing your problem-solving session by attributing a who, what and when to the actions and solutions you have decided upon. The resulting matrix is a simple, easy-to-follow way of ensuring your team can move forward.

Great solutions can’t be enacted without action and ownership. Your problem-solving process should include a stage for allocating tasks to individuals or teams and creating a realistic timeframe for those solutions to be implemented or checked out. Use this method to keep the solution implementation process clear and simple for all involved.

Who/What/When Matrix #gamestorming #action #project planning With Who/What/When matrix, you can connect people with clear actions they have defined and have committed to.

35. Response cards

Group discussion can comprise the bulk of most problem-solving activities and by the end of the process, you might find that your team is talked out!

Providing a means for your team to give feedback with short written notes can ensure everyone is head and can contribute without the need to stand up and talk. Depending on the needs of the group, giving an alternative can help ensure everyone can contribute to your problem-solving model in the way that makes the most sense for them.

Response Cards is a great way to close a workshop if you are looking for a gentle warm-down and want to get some swift discussion around some of the feedback that is raised.

Response Cards #debriefing #closing #structured sharing #questions and answers #thiagi #action It can be hard to involve everyone during a closing of a session. Some might stay in the background or get unheard because of louder participants. However, with the use of Response Cards, everyone will be involved in providing feedback or clarify questions at the end of a session.

Save time and effort discovering the right solutions

A structured problem solving process is a surefire way of solving tough problems, discovering creative solutions and driving organizational change. But how can you design for successful outcomes?

With SessionLab, it’s easy to design engaging workshops that deliver results. Drag, drop and reorder blocks to build your agenda. When you make changes or update your agenda, your session timing adjusts automatically , saving you time on manual adjustments.

Collaborating with stakeholders or clients? Share your agenda with a single click and collaborate in real-time. No more sending documents back and forth over email.

Explore how to use SessionLab to design effective problem solving workshops or watch this five minute video to see the planner in action!

Over to you

The problem-solving process can often be as complicated and multifaceted as the problems they are set-up to solve. With the right problem-solving techniques and a mix of creative exercises designed to guide discussion and generate purposeful ideas, we hope we’ve given you the tools to find the best solutions as simply and easily as possible.

Is there a problem-solving technique that you are missing here? Do you have a favorite activity or method you use when facilitating? Let us know in the comments below, we’d love to hear from you!

thank you very much for these excellent techniques

Certainly wonderful article, very detailed. Shared!

Your list of techniques for problem solving can be helpfully extended by adding TRIZ to the list of techniques. TRIZ has 40 problem solving techniques derived from methods inventros and patent holders used to get new patents. About 10-12 are general approaches. many organization sponsor classes in TRIZ that are used to solve business problems or general organiztational problems. You can take a look at TRIZ and dwonload a free internet booklet to see if you feel it shound be included per your selection process.

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Article • 6 min read

The Reframing Matrix

Using creative perspectives to solve problems.

By the Mind Tools Content Team

When you're stuck on a problem, it often helps to look at it from another perspective. A "fresh pair of eyes" can be all that you need to come up with a great solution.

But sometimes it's difficult to think about what this new perspective might be. You may be so used to dealing with a situation in the same way that you simply can't see any alternative. And, as a result, you may miss out on a better, more efficient, or longer-term answer to your problem.

This is when the Reframing Matrix can help. In this article and in the video, below, we'll look at how you can use this simple tool to look at problems from different points of view.

Click here to view a transcript of this video.

About the Matrix

The Reframing Matrix tool helps you to look at business problems from various perspectives. You can use these perspectives to generate more creative solutions.

The technique was created by Michael Morgan, and published in his 1993 book, "Creating Workforce Innovation." Morgan identified that different people with different experiences will likely approach problems in different ways. The matrix helps you to put yourself into the minds of these people, to imagine the ways that they would face the problem, and to explore the possible solutions that they might suggest. [1]

How to Use the Reframing Matrix

The Reframing Matrix is very easy to use. Here's how to apply the tool in three steps:

Step 1: Draw the Grid

Start by drawing a simple four-square grid, but leave a space in the middle of the grid. Then, in the space, write down the problem that you want to explore.

Figure 1 – Reframing Matrix Step 1

Sometimes, you know there is a business problem that you need to solve, but you're not sure what's causing it. If this applies to you, tools like the Simplex Process and Root Cause Analysis can help you to define your problem clearly.

Step 2: Decide on Perspectives

Now, decide on four different perspectives to use in your matrix. Two useful approaches for doing this are the 4Ps Approach and the Professions Approach .

The 4Ps Approach (not to be confused with the 4Ps of Marketing ) helps you to look at problems from the following perspectives:

Product Perspective: Is there something wrong with the product or service? Is it priced correctly? How well does it serve the market? Is it reliable?
Planning Perspective: Are our business plans, marketing plans, or strategy at fault? Could we improve them?
Potential Perspective: How can we increase sales, or productivity? If we were to seriously increase our targets, or our production volumes, what effect would it have on the problem?
People Perspective: What are the people impacts and people implications of the problem? What do people involved with the problem think? Why are potential customers not using or buying the product?

These examples are just some of the questions that you can ask as you look at your problem using these four perspectives.

The Professions Approach helps you to look at the problem from the viewpoints of different specialists, or stakeholders . For instance, the way a doctor looks at a problem would be different from the approach that a civil engineer or a lawyer would use. Or, the way a CEO views a problem may be different from the way an HR manager sees it.

This approach can be especially useful when you're trying to solve a problem that involves many different types of people, or if you need to step away from your usual way of thinking so that you can be more innovative .

Step 3: Brainstorm Factors

The final step is to brainstorm factors related to your problem from each perspective, and to enter your findings in the appropriate quadrant of the matrix.

If you find yourself slipping into habitual modes of thinking, you could involve other team members in the process, too. Divide them into groups and ask each group to think about the problem from one of the four different perspectives. Or, try collaborating with co-workers from other areas of your organization – their knowledge may help to generate fresh insights.

Once you've completed the matrix, you'll have a better understanding of your problem, and you'll be able to generate more solutions.

The Perceptual Positions and Six Thinking Hats techniques can also be useful when you want to see things from other people's viewpoints.

Our article on CATWOE offers a similar approach. This asks you to look at a problem from the perspectives of Customers, Actors, the Transformation process, the World view, the Owner, and Environmental constraints.

Example Reframing Matrix

In the example in figure 2, below, a manager has used the 4Ps approach to explore why a new product is not selling well.

Figure 2 – Example Reframing Matrix

Example Reframing Matrix adapted from Morgan, M. (1993), 'Creating Workforce Innovation,' West Chatswood, Australia: Business & Professional Publishing, p75, by permission of Allen & Unwin .

The Reframing Matrix tool was created by Michael Morgan, and published in his book, "Creating Workforce Innovation." It helps you to look at a problem from different perspectives.

Use the tool by drawing a simple four-square grid and writing your problem or issue in the middle of the grid.

Then, choose four different perspectives that you will use to look at your problem, and brainstorm factors related to your problem from each of those perspectives.

[1] Morgan, M. (1993). 'Creating Workforce Innovation,' West Chatswood, NSW, Australia: Business & Professional Publishing. p75.

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How to master the seven-step problem-solving process

In this episode of the McKinsey Podcast , Simon London speaks with Charles Conn, CEO of venture-capital firm Oxford Sciences Innovation, and McKinsey senior partner Hugo Sarrazin about the complexities of different problem-solving strategies.

Podcast transcript

Simon London: Hello, and welcome to this episode of the McKinsey Podcast , with me, Simon London. What’s the number-one skill you need to succeed professionally? Salesmanship, perhaps? Or a facility with statistics? Or maybe the ability to communicate crisply and clearly? Many would argue that at the very top of the list comes problem solving: that is, the ability to think through and come up with an optimal course of action to address any complex challenge—in business, in public policy, or indeed in life.

Looked at this way, it’s no surprise that McKinsey takes problem solving very seriously, testing for it during the recruiting process and then honing it, in McKinsey consultants, through immersion in a structured seven-step method. To discuss the art of problem solving, I sat down in California with McKinsey senior partner Hugo Sarrazin and also with Charles Conn. Charles is a former McKinsey partner, entrepreneur, executive, and coauthor of the book Bulletproof Problem Solving: The One Skill That Changes Everything [John Wiley & Sons, 2018].

Charles and Hugo, welcome to the podcast. Thank you for being here.

Hugo Sarrazin: Our pleasure.

Charles Conn: It’s terrific to be here.

Simon London: Problem solving is a really interesting piece of terminology. It could mean so many different things. I have a son who’s a teenage climber. They talk about solving problems. Climbing is problem solving. Charles, when you talk about problem solving, what are you talking about?

Charles Conn: For me, problem solving is the answer to the question “What should I do?” It’s interesting when there’s uncertainty and complexity, and when it’s meaningful because there are consequences. Your son’s climbing is a perfect example. There are consequences, and it’s complicated, and there’s uncertainty—can he make that grab? I think we can apply that same frame almost at any level. You can think about questions like “What town would I like to live in?” or “Should I put solar panels on my roof?”

You might think that’s a funny thing to apply problem solving to, but in my mind it’s not fundamentally different from business problem solving, which answers the question “What should my strategy be?” Or problem solving at the policy level: “How do we combat climate change?” “Should I support the local school bond?” I think these are all part and parcel of the same type of question, “What should I do?”

I’m a big fan of structured problem solving. By following steps, we can more clearly understand what problem it is we’re solving, what are the components of the problem that we’re solving, which components are the most important ones for us to pay attention to, which analytic techniques we should apply to those, and how we can synthesize what we’ve learned back into a compelling story. That’s all it is, at its heart.

I think sometimes when people think about seven steps, they assume that there’s a rigidity to this. That’s not it at all. It’s actually to give you the scope for creativity, which often doesn’t exist when your problem solving is muddled.

Simon London: You were just talking about the seven-step process. That’s what’s written down in the book, but it’s a very McKinsey process as well. Without getting too deep into the weeds, let’s go through the steps, one by one. You were just talking about problem definition as being a particularly important thing to get right first. That’s the first step. Hugo, tell us about that.

Hugo Sarrazin: It is surprising how often people jump past this step and make a bunch of assumptions. The most powerful thing is to step back and ask the basic questions—“What are we trying to solve? What are the constraints that exist? What are the dependencies?” Let’s make those explicit and really push the thinking and defining. At McKinsey, we spend an enormous amount of time in writing that little statement, and the statement, if you’re a logic purist, is great. You debate. “Is it an ‘or’? Is it an ‘and’? What’s the action verb?” Because all these specific words help you get to the heart of what matters.

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Simon London: So this is a concise problem statement.

Hugo Sarrazin: Yeah. It’s not like “Can we grow in Japan?” That’s interesting, but it is “What, specifically, are we trying to uncover in the growth of a product in Japan? Or a segment in Japan? Or a channel in Japan?” When you spend an enormous amount of time, in the first meeting of the different stakeholders, debating this and having different people put forward what they think the problem definition is, you realize that people have completely different views of why they’re here. That, to me, is the most important step.

Charles Conn: I would agree with that. For me, the problem context is critical. When we understand “What are the forces acting upon your decision maker? How quickly is the answer needed? With what precision is the answer needed? Are there areas that are off limits or areas where we would particularly like to find our solution? Is the decision maker open to exploring other areas?” then you not only become more efficient, and move toward what we call the critical path in problem solving, but you also make it so much more likely that you’re not going to waste your time or your decision maker’s time.

How often do especially bright young people run off with half of the idea about what the problem is and start collecting data and start building models—only to discover that they’ve really gone off half-cocked.

Hugo Sarrazin: Yeah.

Charles Conn: And in the wrong direction.

Simon London: OK. So step one—and there is a real art and a structure to it—is define the problem. Step two, Charles?

Charles Conn: My favorite step is step two, which is to use logic trees to disaggregate the problem. Every problem we’re solving has some complexity and some uncertainty in it. The only way that we can really get our team working on the problem is to take the problem apart into logical pieces.

What we find, of course, is that the way to disaggregate the problem often gives you an insight into the answer to the problem quite quickly. I love to do two or three different cuts at it, each one giving a bit of a different insight into what might be going wrong. By doing sensible disaggregations, using logic trees, we can figure out which parts of the problem we should be looking at, and we can assign those different parts to team members.

Simon London: What’s a good example of a logic tree on a sort of ratable problem?

Charles Conn: Maybe the easiest one is the classic profit tree. Almost in every business that I would take a look at, I would start with a profit or return-on-assets tree. In its simplest form, you have the components of revenue, which are price and quantity, and the components of cost, which are cost and quantity. Each of those can be broken out. Cost can be broken into variable cost and fixed cost. The components of price can be broken into what your pricing scheme is. That simple tree often provides insight into what’s going on in a business or what the difference is between that business and the competitors.

If we add the leg, which is “What’s the asset base or investment element?”—so profit divided by assets—then we can ask the question “Is the business using its investments sensibly?” whether that’s in stores or in manufacturing or in transportation assets. I hope we can see just how simple this is, even though we’re describing it in words.

When I went to work with Gordon Moore at the Moore Foundation, the problem that he asked us to look at was “How can we save Pacific salmon?” Now, that sounds like an impossible question, but it was amenable to precisely the same type of disaggregation and allowed us to organize what became a 15-year effort to improve the likelihood of good outcomes for Pacific salmon.

Simon London: Now, is there a danger that your logic tree can be impossibly large? This, I think, brings us onto the third step in the process, which is that you have to prioritize.

Charles Conn: Absolutely. The third step, which we also emphasize, along with good problem definition, is rigorous prioritization—we ask the questions “How important is this lever or this branch of the tree in the overall outcome that we seek to achieve? How much can I move that lever?” Obviously, we try and focus our efforts on ones that have a big impact on the problem and the ones that we have the ability to change. With salmon, ocean conditions turned out to be a big lever, but not one that we could adjust. We focused our attention on fish habitats and fish-harvesting practices, which were big levers that we could affect.

People spend a lot of time arguing about branches that are either not important or that none of us can change. We see it in the public square. When we deal with questions at the policy level—“Should you support the death penalty?” “How do we affect climate change?” “How can we uncover the causes and address homelessness?”—it’s even more important that we’re focusing on levers that are big and movable.

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Simon London: Let’s move swiftly on to step four. You’ve defined your problem, you disaggregate it, you prioritize where you want to analyze—what you want to really look at hard. Then you got to the work plan. Now, what does that mean in practice?

Hugo Sarrazin: Depending on what you’ve prioritized, there are many things you could do. It could be breaking the work among the team members so that people have a clear piece of the work to do. It could be defining the specific analyses that need to get done and executed, and being clear on time lines. There’s always a level-one answer, there’s a level-two answer, there’s a level-three answer. Without being too flippant, I can solve any problem during a good dinner with wine. It won’t have a whole lot of backing.

Simon London: Not going to have a lot of depth to it.

Hugo Sarrazin: No, but it may be useful as a starting point. If the stakes are not that high, that could be OK. If it’s really high stakes, you may need level three and have the whole model validated in three different ways. You need to find a work plan that reflects the level of precision, the time frame you have, and the stakeholders you need to bring along in the exercise.

Charles Conn: I love the way you’ve described that, because, again, some people think of problem solving as a linear thing, but of course what’s critical is that it’s iterative. As you say, you can solve the problem in one day or even one hour.

Charles Conn: We encourage our teams everywhere to do that. We call it the one-day answer or the one-hour answer. In work planning, we’re always iterating. Every time you see a 50-page work plan that stretches out to three months, you know it’s wrong. It will be outmoded very quickly by that learning process that you described. Iterative problem solving is a critical part of this. Sometimes, people think work planning sounds dull, but it isn’t. It’s how we know what’s expected of us and when we need to deliver it and how we’re progressing toward the answer. It’s also the place where we can deal with biases. Bias is a feature of every human decision-making process. If we design our team interactions intelligently, we can avoid the worst sort of biases.

Simon London: Here we’re talking about cognitive biases primarily, right? It’s not that I’m biased against you because of your accent or something. These are the cognitive biases that behavioral sciences have shown we all carry around, things like anchoring, overoptimism—these kinds of things.

Both: Yeah.

Charles Conn: Availability bias is the one that I’m always alert to. You think you’ve seen the problem before, and therefore what’s available is your previous conception of it—and we have to be most careful about that. In any human setting, we also have to be careful about biases that are based on hierarchies, sometimes called sunflower bias. I’m sure, Hugo, with your teams, you make sure that the youngest team members speak first. Not the oldest team members, because it’s easy for people to look at who’s senior and alter their own creative approaches.

Hugo Sarrazin: It’s helpful, at that moment—if someone is asserting a point of view—to ask the question “This was true in what context?” You’re trying to apply something that worked in one context to a different one. That can be deadly if the context has changed, and that’s why organizations struggle to change. You promote all these people because they did something that worked well in the past, and then there’s a disruption in the industry, and they keep doing what got them promoted even though the context has changed.

Simon London: Right. Right.

Hugo Sarrazin: So it’s the same thing in problem solving.

Charles Conn: And it’s why diversity in our teams is so important. It’s one of the best things about the world that we’re in now. We’re likely to have people from different socioeconomic, ethnic, and national backgrounds, each of whom sees problems from a slightly different perspective. It is therefore much more likely that the team will uncover a truly creative and clever approach to problem solving.

Simon London: Let’s move on to step five. You’ve done your work plan. Now you’ve actually got to do the analysis. The thing that strikes me here is that the range of tools that we have at our disposal now, of course, is just huge, particularly with advances in computation, advanced analytics. There’s so many things that you can apply here. Just talk about the analysis stage. How do you pick the right tools?

Charles Conn: For me, the most important thing is that we start with simple heuristics and explanatory statistics before we go off and use the big-gun tools. We need to understand the shape and scope of our problem before we start applying these massive and complex analytical approaches.

Simon London: Would you agree with that?

Hugo Sarrazin: I agree. I think there are so many wonderful heuristics. You need to start there before you go deep into the modeling exercise. There’s an interesting dynamic that’s happening, though. In some cases, for some types of problems, it is even better to set yourself up to maximize your learning. Your problem-solving methodology is test and learn, test and learn, test and learn, and iterate. That is a heuristic in itself, the A/B testing that is used in many parts of the world. So that’s a problem-solving methodology. It’s nothing different. It just uses technology and feedback loops in a fast way. The other one is exploratory data analysis. When you’re dealing with a large-scale problem, and there’s so much data, I can get to the heuristics that Charles was talking about through very clever visualization of data.

You test with your data. You need to set up an environment to do so, but don’t get caught up in neural-network modeling immediately. You’re testing, you’re checking—“Is the data right? Is it sound? Does it make sense?”—before you launch too far.

Simon London: You do hear these ideas—that if you have a big enough data set and enough algorithms, they’re going to find things that you just wouldn’t have spotted, find solutions that maybe you wouldn’t have thought of. Does machine learning sort of revolutionize the problem-solving process? Or are these actually just other tools in the toolbox for structured problem solving?

Charles Conn: It can be revolutionary. There are some areas in which the pattern recognition of large data sets and good algorithms can help us see things that we otherwise couldn’t see. But I do think it’s terribly important we don’t think that this particular technique is a substitute for superb problem solving, starting with good problem definition. Many people use machine learning without understanding algorithms that themselves can have biases built into them. Just as 20 years ago, when we were doing statistical analysis, we knew that we needed good model definition, we still need a good understanding of our algorithms and really good problem definition before we launch off into big data sets and unknown algorithms.

Simon London: Step six. You’ve done your analysis.

Charles Conn: I take six and seven together, and this is the place where young problem solvers often make a mistake. They’ve got their analysis, and they assume that’s the answer, and of course it isn’t the answer. The ability to synthesize the pieces that came out of the analysis and begin to weave those into a story that helps people answer the question “What should I do?” This is back to where we started. If we can’t synthesize, and we can’t tell a story, then our decision maker can’t find the answer to “What should I do?”

Simon London: But, again, these final steps are about motivating people to action, right?

Charles Conn: Yeah.

Simon London: I am slightly torn about the nomenclature of problem solving because it’s on paper, right? Until you motivate people to action, you actually haven’t solved anything.

Charles Conn: I love this question because I think decision-making theory, without a bias to action, is a waste of time. Everything in how I approach this is to help people take action that makes the world better.

Simon London: Hence, these are absolutely critical steps. If you don’t do this well, you’ve just got a bunch of analysis.

Charles Conn: We end up in exactly the same place where we started, which is people speaking across each other, past each other in the public square, rather than actually working together, shoulder to shoulder, to crack these important problems.

Simon London: In the real world, we have a lot of uncertainty—arguably, increasing uncertainty. How do good problem solvers deal with that?

Hugo Sarrazin: At every step of the process. In the problem definition, when you’re defining the context, you need to understand those sources of uncertainty and whether they’re important or not important. It becomes important in the definition of the tree.

You need to think carefully about the branches of the tree that are more certain and less certain as you define them. They don’t have equal weight just because they’ve got equal space on the page. Then, when you’re prioritizing, your prioritization approach may put more emphasis on things that have low probability but huge impact—or, vice versa, may put a lot of priority on things that are very likely and, hopefully, have a reasonable impact. You can introduce that along the way. When you come back to the synthesis, you just need to be nuanced about what you’re understanding, the likelihood.

Often, people lack humility in the way they make their recommendations: “This is the answer.” They’re very precise, and I think we would all be well-served to say, “This is a likely answer under the following sets of conditions” and then make the level of uncertainty clearer, if that is appropriate. It doesn’t mean you’re always in the gray zone; it doesn’t mean you don’t have a point of view. It just means that you can be explicit about the certainty of your answer when you make that recommendation.

Simon London: So it sounds like there is an underlying principle: “Acknowledge and embrace the uncertainty. Don’t pretend that it isn’t there. Be very clear about what the uncertainties are up front, and then build that into every step of the process.”

Hugo Sarrazin: Every step of the process.

Simon London: Yeah. We have just walked through a particular structured methodology for problem solving. But, of course, this is not the only structured methodology for problem solving. One that is also very well-known is design thinking, which comes at things very differently. So, Hugo, I know you have worked with a lot of designers. Just give us a very quick summary. Design thinking—what is it, and how does it relate?

Hugo Sarrazin: It starts with an incredible amount of empathy for the user and uses that to define the problem. It does pause and go out in the wild and spend an enormous amount of time seeing how people interact with objects, seeing the experience they’re getting, seeing the pain points or joy—and uses that to infer and define the problem.

Simon London: Problem definition, but out in the world.

Hugo Sarrazin: With an enormous amount of empathy. There’s a huge emphasis on empathy. Traditional, more classic problem solving is you define the problem based on an understanding of the situation. This one almost presupposes that we don’t know the problem until we go see it. The second thing is you need to come up with multiple scenarios or answers or ideas or concepts, and there’s a lot of divergent thinking initially. That’s slightly different, versus the prioritization, but not for long. Eventually, you need to kind of say, “OK, I’m going to converge again.” Then you go and you bring things back to the customer and get feedback and iterate. Then you rinse and repeat, rinse and repeat. There’s a lot of tactile building, along the way, of prototypes and things like that. It’s very iterative.

Simon London: So, Charles, are these complements or are these alternatives?

Charles Conn: I think they’re entirely complementary, and I think Hugo’s description is perfect. When we do problem definition well in classic problem solving, we are demonstrating the kind of empathy, at the very beginning of our problem, that design thinking asks us to approach. When we ideate—and that’s very similar to the disaggregation, prioritization, and work-planning steps—we do precisely the same thing, and often we use contrasting teams, so that we do have divergent thinking. The best teams allow divergent thinking to bump them off whatever their initial biases in problem solving are. For me, design thinking gives us a constant reminder of creativity, empathy, and the tactile nature of problem solving, but it’s absolutely complementary, not alternative.

Simon London: I think, in a world of cross-functional teams, an interesting question is do people with design-thinking backgrounds really work well together with classical problem solvers? How do you make that chemistry happen?

Hugo Sarrazin: Yeah, it is not easy when people have spent an enormous amount of time seeped in design thinking or user-centric design, whichever word you want to use. If the person who’s applying classic problem-solving methodology is very rigid and mechanical in the way they’re doing it, there could be an enormous amount of tension. If there’s not clarity in the role and not clarity in the process, I think having the two together can be, sometimes, problematic.

The second thing that happens often is that the artifacts the two methodologies try to gravitate toward can be different. Classic problem solving often gravitates toward a model; design thinking migrates toward a prototype. Rather than writing a big deck with all my supporting evidence, they’ll bring an example, a thing, and that feels different. Then you spend your time differently to achieve those two end products, so that’s another source of friction.

Now, I still think it can be an incredibly powerful thing to have the two—if there are the right people with the right mind-set, if there is a team that is explicit about the roles, if we’re clear about the kind of outcomes we are attempting to bring forward. There’s an enormous amount of collaborativeness and respect.

Simon London: But they have to respect each other’s methodology and be prepared to flex, maybe, a little bit, in how this process is going to work.

Hugo Sarrazin: Absolutely.

Simon London: The other area where, it strikes me, there could be a little bit of a different sort of friction is this whole concept of the day-one answer, which is what we were just talking about in classical problem solving. Now, you know that this is probably not going to be your final answer, but that’s how you begin to structure the problem. Whereas I would imagine your design thinkers—no, they’re going off to do their ethnographic research and get out into the field, potentially for a long time, before they come back with at least an initial hypothesis.

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Hugo Sarrazin: That is a great callout, and that’s another difference. Designers typically will like to soak into the situation and avoid converging too quickly. There’s optionality and exploring different options. There’s a strong belief that keeps the solution space wide enough that you can come up with more radical ideas. If there’s a large design team or many designers on the team, and you come on Friday and say, “What’s our week-one answer?” they’re going to struggle. They’re not going to be comfortable, naturally, to give that answer. It doesn’t mean they don’t have an answer; it’s just not where they are in their thinking process.

Simon London: I think we are, sadly, out of time for today. But Charles and Hugo, thank you so much.

Charles Conn: It was a pleasure to be here, Simon.

Hugo Sarrazin: It was a pleasure. Thank you.

Simon London: And thanks, as always, to you, our listeners, for tuning into this episode of the McKinsey Podcast . If you want to learn more about problem solving, you can find the book, Bulletproof Problem Solving: The One Skill That Changes Everything , online or order it through your local bookstore. To learn more about McKinsey, you can of course find us at McKinsey.com.

Charles Conn is CEO of Oxford Sciences Innovation and an alumnus of McKinsey’s Sydney office. Hugo Sarrazin is a senior partner in the Silicon Valley office, where Simon London, a member of McKinsey Publishing, is also based.

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4.6: Solve Systems of Equations Using Matrices

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Learning Objectives

By the end of this section, you will be able to:

Write the augmented matrix for a system of equations
Use row operations on a matrix
Solve systems of equations using matrices

Before you get started, take this readiness quiz.

Solve: $3(x+2)+4=4(2x−1)+9$. If you missed this problem, review [link] .
Solve: $0.25p+0.25(x+4)=5.20$. If you missed this problem, review [link] .
Evaluate when $x=−2$ and $y=3:2x^2−xy+3y^2$. If you missed this problem, review [link] .

Write the Augmented Matrix for a System of Equations

Solving a system of equations can be a tedious operation where a simple mistake can wreak havoc on finding the solution. An alternative method which uses the basic procedures of elimination but with notation that is simpler is available. The method involves using a matrix . A matrix is a rectangular array of numbers arranged in rows and columns.

A matrix is a rectangular array of numbers arranged in rows and columns.

A matrix with m rows and n columns has order $m\times n$. The matrix on the left below has 2 rows and 3 columns and so it has order $2\times 3$. We say it is a 2 by 3 matrix.

$Figure shows two matrices. The one on the left has the numbers minus 3, minus 2 and 2 in the first row and the numbers minus 1, 4 and 5 in the second row. The rows and columns are enclosed within brackets. Thus, it has 2 rows and 3 columns. It is labeled 2 cross 3 or 2 by 3 matrix. The matrix on the right is similar but with 3 rows and 4 columns. It is labeled 3 by 4 matrix.$

Each number in the matrix is called an element or entry in the matrix.

We will use a matrix to represent a system of linear equations. We write each equation in standard form and the coefficients of the variables and the constant of each equation becomes a row in the matrix. Each column then would be the coefficients of one of the variables in the system or the constants. A vertical line replaces the equal signs. We call the resulting matrix the augmented matrix for the system of equations.

$The equations are 3x plus y equals minus 3 and 2x plus 3y equals 6. A 2 by 3 matrix is shown. The first row is 3, 1, minus 3. The second row is 2, 3, 6. The first column is labeled coefficients of x. The second column is labeled coefficients of y and the third is labeled constants.$

Notice the first column is made up of all the coefficients of x , the second column is the all the coefficients of y , and the third column is all the constants.

Example $\PageIndex{1}$

ⓐ $\left\{ \begin{array} {l} 5x−3y=−1 \\ y=2x−2 \end{array} \right. $ ⓑ $ \left\{ \begin{array} {l} 6x−5y+2z=3 \\ 2x+y−4z=5 \\ 3x−3y+z=−1 \end{array} \right. $

ⓐ The second equation is not in standard form. We rewrite the second equation in standard form.

\[\begin{aligned} y=2x−2 \\ −2x+y=−2 \end{aligned} \nonumber\]

We replace the second equation with its standard form. In the augmented matrix, the first equation gives us the first row and the second equation gives us the second row. The vertical line replaces the equal signs.

ⓑ All three equations are in standard form. In the augmented matrix the first equation gives us the first row, the second equation gives us the second row, and the third equation gives us the third row. The vertical line replaces the equal signs.

$The equations are 6x minus 5y plus 2z equals 3, 2x plus y minus 4z equals 5 and 3x minus 3y plus z equals minus 1. A 4 by 3 matrix is shown whose first row is 6, minus 5, 2, 3. Its second row is 2, 1, minus 4, 5. Its third row is 3, minus 3, 1 and minus 1. Its first three columns are labeled x, y and z respectively.$

Example $\PageIndex{2}$

Write each system of linear equations as an augmented matrix:

ⓐ $\left\{ \begin{array} {l} 3x+8y=−3 \\ 2x=−5y−3 \end{array} \right. $ ⓑ $\left\{ \begin{array} {l} 2x−5y+3z=8 \\ 3x−y+4z=7 \\ x+3y+2z=−3 \end{array} \right. $

ⓐ $\left[ \begin{matrix} 3 &8 &-3 \\ 2 &5 &−3 \end{matrix} \right] $

ⓑ $\left[ \begin{matrix} 2 &3 &1 &−5 \\ −1 &3 &3 &4 \\ 2 &8 &7 &−3 \end{matrix} \right] $

Example $\PageIndex{3}$

ⓐ $\left\{ \begin{array} {l} 11x=−9y−5 \\ 7x+5y=−1 \end{array} \right. $ ⓑ $\left\{ \begin{array} {l} 5x−3y+2z=−5 \\ 2x−y−z=4 \\ 3x−2y+2z=−7 \end{array} \right. $

ⓐ $\left[ \begin{matrix} 11 &9 &−5 \\ 7 &5 &−1 \end{matrix} \right] $ ⓑ $\left[ \begin{matrix} 5 &−3 &2 &−5 \\ 2 &−1 &−1 &4 \\ 3 &−2 &2 &−7 \end{matrix} \right] $

It is important as we solve systems of equations using matrices to be able to go back and forth between the system and the matrix. The next example asks us to take the information in the matrix and write the system of equations.

Example $\PageIndex{4}$

Write the system of equations that corresponds to the augmented matrix:

$\left[ \begin{array} {ccc|c} 4 &−3 &3 &−1 \\ 1 &2 &−1 &2 \\ −2 &−1 &3 &−4 \end{array} \right] $.

We remember that each row corresponds to an equation and that each entry is a coefficient of a variable or the constant. The vertical line replaces the equal sign. Since this matrix is a $4\times 3$, we know it will translate into a system of three equations with three variables.

$A 3 by 4 matrix is shown. Its first row is 4, minus 3, 3, minus 1. Its second row is 1, 2, minus 1, 2. Its third row is minus 2, minus 1, 3, minus 4. The three equations are 4x minus 3y plus 3z equals minus 1, x plus 2y minus z equals 2 and minus 2x minus y plus 3z equals minus 4.$

Example $\PageIndex{5}$

Write the system of equations that corresponds to the augmented matrix: $\left[ \begin{matrix} 1 &−1 &2 &3 \\ 2 &1 &−2 &1 \\ 4 &−1 &2 &0 \end{matrix} \right] $.

$\left\{ \begin{array} {l} x−y+2z=3 \\ 2x+y−2z=1 \\ 4x−y+2z=0 \end{array} \right.$

Example $\PageIndex{6}$

Write the system of equations that corresponds to the augmented matrix: $\left[ \begin{matrix} 1 &1 &1 &4 \\ 2 &3 &−1 &8 \\ 1 &1 &−1 &3 \end{matrix} \right] $.

$\left\{ \begin{array} {l} x+y+z=4 \\ 2x+3y−z=8 \\ x+y−z=3 \end{array} \right.$

Use Row Operations on a Matrix

Once a system of equations is in its augmented matrix form, we will perform operations on the rows that will lead us to the solution.

To solve by elimination, it doesn’t matter which order we place the equations in the system. Similarly, in the matrix we can interchange the rows.

When we solve by elimination, we often multiply one of the equations by a constant. Since each row represents an equation, and we can multiply each side of an equation by a constant, similarly we can multiply each entry in a row by any real number except 0.

In elimination, we often add a multiple of one row to another row. In the matrix we can replace a row with its sum with a multiple of another row.

These actions are called row operations and will help us use the matrix to solve a system of equations.

ROW OPERATIONS

In a matrix, the following operations can be performed on any row and the resulting matrix will be equivalent to the original matrix.

Interchange any two rows.
Multiply a row by any real number except 0.
Add a nonzero multiple of one row to another row.

Performing these operations is easy to do but all the arithmetic can result in a mistake. If we use a system to record the row operation in each step, it is much easier to go back and check our work.

We use capital letters with subscripts to represent each row. We then show the operation to the left of the new matrix. To show interchanging a row:

$A 2 by 3 matrix is shown. Its first row, labeled R2 is 2, minus 1, 2. Its second row, labeled R1 is 5, minus 3, minus 1.$

To multiply row 2 by $−3$:

$A 2 by 3 matrix is shown. Its first row is 5, minus 3, minus 1. Its second row is 2, minus 1, 2. An arrow point from this matrix to another one on the right. The first row of the new matrix is the same. The second row is preceded by minus 3 R2. It is minus 6, 3, minus 6.$

To multiply row 2 by $−3$ and add it to row 1:

$A 2 by 3 matrix is shown. Its first row is 5, minus 3, minus 1. Its second row is 2, minus 1, 2. An arrow point from this matrix to another one on the right. The first row of the new matrix is preceded by minus 3 R2 plus R1. It is minus 1, 0, minus 7. The second row is 2, minus 1, 2.$

Example $\PageIndex{7}$

Perform the indicated operations on the augmented matrix:

ⓐ Interchange rows 2 and 3.

ⓑ Multiply row 2 by 5.

$ \left[ \begin{array} {ccc|c} 6 &−5 &2 &3 \\ 2 &1 &−4 &5 \\ 3 &−3 &1 &−1 \end{array} \right] $

ⓐ We interchange rows 2 and 3.

$Two 3 by 4 matrices are shown. In the one on the left, the first row is 6, minus 5, 2, 3. The second row is 2, 1, minus 4, 5. The third row is 3, minus 3, 1, minus 1. The second matrix is similar except that rows 2 and 3 are interchanged.$

ⓑ We multiply row 2 by 5.

$In the 3 by 4 matrix, the first row is 6, minus 5, 2, 3. The second row is 2, 1, minus 4, 5. The third row is 3, minus 3, 1, minus 1. Performing the operation minus 2 R3 plus R1 on the first row, the first row becomes 6 plus minus 2 times 3, minus 5 plus minus 2 times minus 3, 2 plus minus 2 times 1 and 3 plus minus 2 times minus 1. This becomes 0, 1, 0, 5. The remaining 2 rows of the new matrix are the same.$

Example $\PageIndex{8}$

ⓐ Interchange rows 1 and 3.

ⓑ Multiply row 3 by 3.

$ \left[ \begin{array} {ccc|c} 5 &−2 &-2 &-2 \\ 4 &-1 &−4 &4 \\ -2 &3 &0 &−1 \end{array} \right] $

ⓐ $ \left[ \begin{matrix} −2 &3 &0 &−2 \\ 4 &−1 &−4 &4 \\ 5 &−2 &−2 &−2 \end{matrix} \right] $

ⓑ $ \left[ \begin{matrix} −2 &3 &0 &−2 \\ 4 &−1 &−4 &4 \\ 15 &−6 &−6 &−6 \end{matrix} \right] $

Example $\PageIndex{9}$

ⓐ Interchange rows 1 and 2,

ⓑ Multiply row 1 by 2,

$ \left[ \begin{array} {ccc|c} 2 &−3 &−2 &−4 \\ 4 &1 &−3 &2 \\ 5 &0 &4 &−1 \end{array} \right] $

ⓐ $ \left[ \begin{matrix} 4 &1 &−3 &2 \\ 2 &−3 &−2 &−4 \\ 5 &0 &4 &−1 \end{matrix} \right] $ ⓑ $ \left[ \begin{matrix} 8 &2 &−6 &4 \\ 2 &−3 &−2 &−4 \\ 5 &0 &4 &−1 \end{matrix} \right] $ ⓒ $ \left[ \begin{matrix} 14 &−7 &−12 &−8 \\ 2 &−3 &−2 &−4 \\ 5 &0 &4 &−1 \end{matrix} \right] $

Now that we have practiced the row operations, we will look at an augmented matrix and figure out what operation we will use to reach a goal. This is exactly what we did when we did elimination. We decided what number to multiply a row by in order that a variable would be eliminated when we added the rows together.

Given this system, what would you do to eliminate x ?

$The two equations are x minus y equals 2 and 4x minus 8y equals 0. Multiplying the first by minus 4, we get minus 4x plus 4y equals minus 8. Adding this to the second equation we get minus 4y equals minus 8.$

This next example essentially does the same thing, but to the matrix.

Example $\PageIndex{10}$

Perform the needed row operation that will get the first entry in row 2 to be zero in the augmented matrix: $ \left[ \begin{array} {cc|c} 1 &−1 &2 \\ 4 &−8 &0 \end{array} \right] $

To make the 4 a 0, we could multiply row 1 by $−4$ and then add it to row 2.

$The 2 by 3 matrix is 1, minus 1, 2 and 4, minus 8, 0. Performing the operation minus 4R1 plus R2 on row 2, the second row of the new matrix becomes 0, minus 4, minus 8. The first row remains the same.$

Example $\PageIndex{11}$

Perform the needed row operation that will get the first entry in row 2 to be zero in the augmented matrix: $ \left[ \begin{array} {cc|c} 1 &−1 &2 \\ 3 &−6 &2 \end{array} \right] $

$ \left[ \begin{matrix} 1 &−1 &2 \\ 0 &−3 &−4 \end{matrix} \right] $

Example $\PageIndex{12}$

Perform the needed row operation that will get the first entry in row 2 to be zero in the augmented matrix: $ \left[ \begin{array} {cc|c} 1 &−1 &3 \\ -2 &−3 &2 \end{array} \right] $

$ \left[ \begin{matrix} 1 &−1 &3 \\ 0 &−5 &8 \end{matrix} \right] $

Solve Systems of Equations Using Matrices

To solve a system of equations using matrices, we transform the augmented matrix into a matrix in row-echelon form using row operations. For a consistent and independent system of equations, its augmented matrix is in row-echelon form when to the left of the vertical line, each entry on the diagonal is a 1 and all entries below the diagonal are zeros.

ROW-ECHELON FORM

For a consistent and independent system of equations, its augmented matrix is in row-echelon form when to the left of the vertical line, each entry on the diagonal is a 1 and all entries below the diagonal are zeros.

$A 2 by 3 matrix is shown on the left. Its first row is 1, a, b. Its second row is 0, 1, c. An arrow points diagonally down and right, overlapping both the 1s in the matrix. A 3 by 4 matrix is shown on the right. Its first row is 1, a, b, d. Its second row is 0, 1, c, e. Its third row is 0, 0, 1, f. An arrow points diagonally down and right, overlapping all the 1s in the matrix. a, b, c, d, e, f are real numbers.$

Once we get the augmented matrix into row-echelon form, we can write the equivalent system of equations and read the value of at least one variable. We then substitute this value in another equation to continue to solve for the other variables. This process is illustrated in the next example.

How to Solve a System of Equations Using a Matrix

Solve the system of equations using a matrix: $\left\{ \begin{array} {l} 3x+4y=5 \\ x+2y=1 \end{array} \right. $

$The equations are 3x plus 4y equals 5 and x plus 2y equals 1. Step 1. Write the augmented matrix for the system of equations. We get a 2 by 3 matrix with first row 3, 4, 5 and second row 1, 2, 1.$

Example $\PageIndex{14}$

Solve the system of equations using a matrix: $\left\{ \begin{array} {l} 2x+y=7 \\ x−2y=6 \end{array} \right. $

The solution is $(4,−1)$.

Example $\PageIndex{15}$

Solve the system of equations using a matrix: $\left\{ \begin{array} {l} 2x+y=−4 \\ x−y=−2 \end{array} \right. $

The solution is $(−2,0)$.

The steps are summarized here.

SOLVE A SYSTEM OF EQUATIONS USING MATRICES.

Write the augmented matrix for the system of equations.
Using row operations get the entry in row 1, column 1 to be 1.
Using row operations, get zeros in column 1 below the 1.
Using row operations, get the entry in row 2, column 2 to be 1.
Continue the process until the matrix is in row-echelon form.
Write the corresponding system of equations.
Use substitution to find the remaining variables.
Write the solution as an ordered pair or triple.
Check that the solution makes the original equations true.

Here is a visual to show the order for getting the 1’s and 0’s in the proper position for row-echelon form.

$The figure shows 3 steps for a 2 by 3 matrix and 6 steps for a 3 by 4 matrix. For the former, step 1 is to get a 1 in row 1 column 1. Step to is to get a 0 is row 2 column 1. Step 3 is to get a 1 in row 2 column 2. For a 3 by 4 matrix, step 1 is to get a 1 in row 1 column 1. Step 2 is to get a 0 in row 2 column 1. Step 3 is to get a 0 in row 3 column 1. Step 4 is to get a 1 in row 2 column 2. Step 5 is to get a 0 in row 3 column 2. Step 6 is to get a 1 in row 3 column 3.$

We use the same procedure when the system of equations has three equations.

Example $\PageIndex{16}$

Solve the system of equations using a matrix: $\left\{ \begin{array} {l} 3x+8y+2z=−5 \\ 2x+5y−3z=0 \\ x+2y−2z=−1 \end{array} \right. $

Example $\PageIndex{17}$

Solve the system of equations using a matrix: $\left\{ \begin{array} {l} 2x−5y+3z=8 \\ 3x−y+4z=7 \\ x+3y+2z=−3 \end{array} \right. $

$(6,−1,−3)$

Example $\PageIndex{18}$

Solve the system of equations using a matrix: $\left\{ \begin{array} {l} −3x+y+z=−4 \\ −x+2y−2z=1 \\ 2x−y−z=−1 \end{array} \right. $

$(5,7,4)$

So far our work with matrices has only been with systems that are consistent and independent, which means they have exactly one solution. Let’s now look at what happens when we use a matrix for a dependent or inconsistent system.

Example $\PageIndex{19}$

Solve the system of equations using a matrix: $\left\{ \begin{array} {l} x+y+3z=0 \\ x+3y+5z=0 \\ 2x+4z=1 \end{array} \right. $

Example $\PageIndex{20}$

Solve the system of equations using a matrix: $\left\{ \begin{array} {l} x−2y+2z=1 \\ −2x+y−z=2 \\ x−y+z=5 \end{array} \right. $

no solution

Example $\PageIndex{21}$

Solve the system of equations using a matrix: $\left\{ \begin{array} {l} 3x+4y−3z=−2 \\ −2x+3y−z=−1 \\ 2x+y−2z=6 \end{array} \right. $

The last system was inconsistent and so had no solutions. The next example is dependent and has infinitely many solutions.

Example $\PageIndex{22}$

Solve the system of equations using a matrix: $\left\{ \begin{array} {l} x−2y+3z=1 \\ x+y−3z=7 \\ 3x−4y+5z=7 \end{array} \right. $

Example $\PageIndex{23}$

Solve the system of equations using a matrix: $\left\{ \begin{array} {l} x+y−z=0 \\ 2x+4y−2z=6 \\ 3x+6y−3z=9 \end{array} \right. $

infinitely many solutions $(x,y,z)$, where $x=z−3;\space y=3;\space z$ is any real number.

Example $\PageIndex{24}$

Solve the system of equations using a matrix: $\left\{ \begin{array} {l} x−y−z=1 \\ −x+2y−3z=−4 \\ 3x−2y−7z=0 \end{array} \right. $

infinitely many solutions $(x,y,z)$, where $x=5z−2;\space y=4z−3;\space z$ is any real number.

Access this online resource for additional instruction and practice with Gaussian Elimination.

Gaussian Elimination

Key Concepts

Interchange any two rows
Multiply a row by any real number except 0
Add a nonzero multiple of one row to another row

7.5 Matrices and Matrix Operations

Learning objectives.

In this section, you will:

Find the sum and difference of two matrices.
Find scalar multiples of a matrix.
Find the product of two matrices.

Two club soccer teams, the Wildcats and the Mud Cats, are hoping to obtain new equipment for an upcoming season. Table 1 shows the needs of both teams.

A goal costs $300; a ball costs $10; and a jersey costs $30. How can we find the total cost for the equipment needed for each team? In this section, we discover a method in which the data in the soccer equipment table can be displayed and used for calculating other information. Then, we will be able to calculate the cost of the equipment.

Finding the Sum and Difference of Two Matrices

To solve a problem like the one described for the soccer teams, we can use a matrix , which is a rectangular array of numbers. A row in a matrix is a set of numbers that are aligned horizontally. A column in a matrix is a set of numbers that are aligned vertically. Each number is an entry , sometimes called an element, of the matrix. Matrices (plural) are enclosed in [ ] or ( ), and are usually named with capital letters. For example, three matrices named A , B , A , B , and C C are shown below.

Describing Matrices

A matrix is often referred to by its size or dimensions: m × n m × n indicating m m rows and n n columns. Matrix entries are defined first by row and then by column. For example, to locate the entry in matrix A A identified as a i j , a i j , we look for the entry in row i , i , column j . j . In matrix A , A , shown below, the entry in row 2, column 3 is a 23 . a 23 .

A square matrix is a matrix with dimensions n × n , n × n , meaning that it has the same number of rows as columns. The 3 × 3 3 × 3 matrix above is an example of a square matrix.

A row matrix is a matrix consisting of one row with dimensions 1 × n . 1 × n .

A column matrix is a matrix consisting of one column with dimensions m × 1. m × 1.

A matrix may be used to represent a system of equations. In these cases, the numbers represent the coefficients of the variables in the system. Matrices often make solving systems of equations easier because they are not encumbered with variables. We will investigate this idea further in the next section, but first we will look at basic matrix operations .

A matrix is a rectangular array of numbers that is usually named by a capital letter: A , B , C , A , B , C , and so on. Each entry in a matrix is referred to as a i j , a i j , such that i i represents the row and j j represents the column. Matrices are often referred to by their dimensions: m × n m × n indicating m m rows and n n columns.

Finding the Dimensions of the Given Matrix and Locating Entries

Given matrix A : A :

ⓐ What are the dimensions of matrix A ? A ?
ⓑ What are the entries at a 31 a 31 and a 22 ? a 22 ? A = [ 2 1 0 2 4 7 3 1 − 2 ] A = [ 2 1 0 2 4 7 3 1 − 2 ]
ⓐ The dimensions are 3 × 3 3 × 3 because there are three rows and three columns.
ⓑ Entry a 31 a 31 is the number at row 3, column 1, which is 3. The entry a 22 a 22 is the number at row 2, column 2, which is 4. Remember, the row comes first, then the column.

Adding and Subtracting Matrices

We use matrices to list data or to represent systems. Because the entries are numbers, we can perform operations on matrices. We add or subtract matrices by adding or subtracting corresponding entries.

In order to do this, the entries must correspond. Therefore, addition and subtraction of matrices is only possible when the matrices have the same dimensions . We can add or subtract a 3 × 3 3 × 3 matrix and another 3 × 3 3 × 3 matrix, but we cannot add or subtract a 2 × 3 2 × 3 matrix and a 3 × 3 3 × 3 matrix because some entries in one matrix will not have a corresponding entry in the other matrix.

Given matrices A A and B B of like dimensions, addition and subtraction of A A and B B will produce matrix C C or matrix D D of the same dimension.

Matrix addition is commutative.

It is also associative.

Finding the Sum of Matrices

Find the sum of A A and B , B , given

Add corresponding entries.

Adding Matrix A and Matrix B

Find the sum of A A and B . B .

Add corresponding entries. Add the entry in row 1, column 1, a 11 , a 11 , of matrix A A to the entry in row 1, column 1, b 11 , b 11 , of B . B . Continue the pattern until all entries have been added.

Finding the Difference of Two Matrices

Find the difference of A A and B . B .

We subtract the corresponding entries of each matrix.

Finding the Sum and Difference of Two 3 x 3 Matrices

Given A A and B : B :

ⓐ Find the sum.
ⓑ Find the difference.
ⓐ Add the corresponding entries. A + B = [ 2 − 10 − 2 14 12 10 4 − 2 2 ] + [ 6 10 − 2 0 − 12 − 4 − 5 2 − 2 ] = [ 2 + 6 − 10 + 10 − 2 − 2 14 + 0 12 − 12 10 − 4 4 − 5 − 2 + 2 2 − 2 ] = [ 8 0 − 4 14 0 6 − 1 0 0 ] A + B = [ 2 − 10 − 2 14 12 10 4 − 2 2 ] + [ 6 10 − 2 0 − 12 − 4 − 5 2 − 2 ] = [ 2 + 6 − 10 + 10 − 2 − 2 14 + 0 12 − 12 10 − 4 4 − 5 − 2 + 2 2 − 2 ] = [ 8 0 − 4 14 0 6 − 1 0 0 ]
ⓑ Subtract the corresponding entries. A − B = [ 2 −10 −2 14 12 10 4 −2 2 ] − [ 6 10 −2 0 −12 −4 −5 2 −2 ] = [ 2 − 6 −10 − 10 −2 + 2 14 − 0 12 + 12 10 + 4 4 + 5 −2 − 2 2 + 2 ] = [ −4 −20 0 14 24 14 9 −4 4 ] A − B = [ 2 −10 −2 14 12 10 4 −2 2 ] − [ 6 10 −2 0 −12 −4 −5 2 −2 ] = [ 2 − 6 −10 − 10 −2 + 2 14 − 0 12 + 12 10 + 4 4 + 5 −2 − 2 2 + 2 ] = [ −4 −20 0 14 24 14 9 −4 4 ]

Add matrix A A and matrix B . B .

Finding Scalar Multiples of a Matrix

Besides adding and subtracting whole matrices, there are many situations in which we need to multiply a matrix by a constant called a scalar. Recall that a scalar is a real number quantity that has magnitude, but not direction. For example, time, temperature, and distance are scalar quantities. The process of scalar multiplication involves multiplying each entry in a matrix by a scalar. A scalar multiple is any entry of a matrix that results from scalar multiplication.

Consider a real-world scenario in which a university needs to add to its inventory of computers, computer tables, and chairs in two of the campus labs due to increased enrollment. They estimate that 15% more equipment is needed in both labs. The school’s current inventory is displayed in Table 2 .

Converting the data to a matrix, we have

To calculate how much computer equipment will be needed, we multiply all entries in matrix C C by 0.15.

We must round up to the next integer, so the amount of new equipment needed is

Adding the two matrices as shown below, we see the new inventory amounts.

Thus, Lab A will have 18 computers, 19 computer tables, and 19 chairs; Lab B will have 32 computers, 40 computer tables, and 40 chairs.

Scalar Multiplication

Scalar multiplication involves finding the product of a constant by each entry in the matrix. Given

the scalar multiple c A c A is

Scalar multiplication is distributive. For the matrices A , B , A , B , and C C with scalars a a and b , b ,

Multiplying the Matrix by a Scalar

Multiply matrix A A by the scalar 3.

Multiply each entry in A A by the scalar 3.

Given matrix B , B , find −2 B −2 B where

Finding the Sum of Scalar Multiples

Find the sum 3 A + 2 B . 3 A + 2 B .

First, find 3 A , 3 A , then 2 B . 2 B .

Now, add 3 A + 2 B . 3 A + 2 B .

Finding the Product of Two Matrices

In addition to multiplying a matrix by a scalar, we can multiply two matrices. Finding the product of two matrices is only possible when the inner dimensions are the same, meaning that the number of columns of the first matrix is equal to the number of rows of the second matrix. If A A is an m × r m × r matrix and B B is an r × n r × n matrix, then the product matrix A B A B is an m × n m × n matrix. For example, the product A B A B is possible because the number of columns in A A is the same as the number of rows in B . B . If the inner dimensions do not match, the product is not defined.

We multiply entries of A A with entries of B B according to a specific pattern as outlined below. The process of matrix multiplication becomes clearer when working a problem with real numbers.

To obtain the entries in row i i of A B , A B , we multiply the entries in row i i of A A by column j j in B B and add. For example, given matrices A A and B , B , where the dimensions of A A are 2 × 3 2 × 3 and the dimensions of B B are 3 × 3 , 3 × 3 , the product of A B A B will be a 2 × 3 2 × 3 matrix.

Multiply and add as follows to obtain the first entry of the product matrix A B . A B .

To obtain the entry in row 1, column 1 of A B , A B , multiply the first row in A A by the first column in B , B , and add. [ a 11 a 12 a 13 ] [ b 11 b 21 b 31 ] = a 11 ⋅ b 11 + a 12 ⋅ b 21 + a 13 ⋅ b 31 [ a 11 a 12 a 13 ] [ b 11 b 21 b 31 ] = a 11 ⋅ b 11 + a 12 ⋅ b 21 + a 13 ⋅ b 31
To obtain the entry in row 1, column 2 of A B , A B , multiply the first row of A A by the second column in B , B , and add. [ a 11 a 12 a 13 ] [ b 12 b 22 b 32 ] = a 11 ⋅ b 12 + a 12 ⋅ b 22 + a 13 ⋅ b 32 [ a 11 a 12 a 13 ] [ b 12 b 22 b 32 ] = a 11 ⋅ b 12 + a 12 ⋅ b 22 + a 13 ⋅ b 32
To obtain the entry in row 1, column 3 of A B , A B , multiply the first row of A A by the third column in B , B , and add. [ a 11 a 12 a 13 ] [ b 13 b 23 b 33 ] = a 11 ⋅ b 13 + a 12 ⋅ b 23 + a 13 ⋅ b 33 [ a 11 a 12 a 13 ] [ b 13 b 23 b 33 ] = a 11 ⋅ b 13 + a 12 ⋅ b 23 + a 13 ⋅ b 33

We proceed the same way to obtain the second row of A B . A B . In other words, row 2 of A A times column 1 of B ; B ; row 2 of A A times column 2 of B ; B ; row 2 of A A times column 3 of B . B . When complete, the product matrix will be

Properties of Matrix Multiplication

For the matrices A , B , A , B , and C C the following properties hold.

Matrix multiplication is associative: ( A B ) C = A ( B C ) . ( A B ) C = A ( B C ) .
Matrix multiplication is distributive: C ( A + B ) = C A + C B , ( A + B ) C = A C + B C . C ( A + B ) = C A + C B , ( A + B ) C = A C + B C .

Note that matrix multiplication is not commutative.

Multiplying Two Matrices

Multiply matrix A A and matrix B . B .

First, we check the dimensions of the matrices. Matrix A A has dimensions 2 × 2 2 × 2 and matrix B B has dimensions 2 × 2. 2 × 2. The inner dimensions are the same so we can perform the multiplication. The product will have the dimensions 2 × 2. 2 × 2.

We perform the operations outlined previously.

ⓐ Find A B . A B .
ⓑ Find B A . B A .
ⓐ As the dimensions of A A are 2 × 3 2 × 3 and the dimensions of B B are 3 × 2 , 3 × 2 , these matrices can be multiplied together because the number of columns in A A matches the number of rows in B . B . The resulting product will be a 2 × 2 2 × 2 matrix, the number of rows in A A by the number of columns in B . B . A B = [ −1 2 3 4 0 5 ] [ 5 −1 − 4 0 2 3 ] = [ −1 ( 5 ) + 2 ( −4 ) + 3 ( 2 ) −1 ( −1 ) + 2 ( 0 ) + 3 ( 3 ) 4 ( 5 ) + 0 ( −4 ) + 5 ( 2 ) 4 ( −1 ) + 0 ( 0 ) + 5 ( 3 ) ] = [ −7 10 30 11 ] A B = [ −1 2 3 4 0 5 ] [ 5 −1 − 4 0 2 3 ] = [ −1 ( 5 ) + 2 ( −4 ) + 3 ( 2 ) −1 ( −1 ) + 2 ( 0 ) + 3 ( 3 ) 4 ( 5 ) + 0 ( −4 ) + 5 ( 2 ) 4 ( −1 ) + 0 ( 0 ) + 5 ( 3 ) ] = [ −7 10 30 11 ]
ⓑ The dimensions of B B are 3 × 2 3 × 2 and the dimensions of A A are 2 × 3. 2 × 3. The inner dimensions match so the product is defined and will be a 3 × 3 3 × 3 matrix. B A = [ 5 −1 −4 0 2 3 ] [ −1 2 3 4 0 5 ] = [ 5 ( −1 ) + −1 ( 4 ) 5 ( 2 ) + −1 ( 0 ) 5 ( 3 ) + −1 ( 5 ) −4 ( −1 ) + 0 ( 4 ) −4 ( 2 ) + 0 ( 0 ) −4 ( 3 ) + 0 ( 5 ) 2 ( −1 ) + 3 ( 4 ) 2 ( 2 ) + 3 ( 0 ) 2 ( 3 ) + 3 ( 5 ) ] = [ −9 10 10 4 −8 −12 10 4 21 ] B A = [ 5 −1 −4 0 2 3 ] [ −1 2 3 4 0 5 ] = [ 5 ( −1 ) + −1 ( 4 ) 5 ( 2 ) + −1 ( 0 ) 5 ( 3 ) + −1 ( 5 ) −4 ( −1 ) + 0 ( 4 ) −4 ( 2 ) + 0 ( 0 ) −4 ( 3 ) + 0 ( 5 ) 2 ( −1 ) + 3 ( 4 ) 2 ( 2 ) + 3 ( 0 ) 2 ( 3 ) + 3 ( 5 ) ] = [ −9 10 10 4 −8 −12 10 4 21 ]

Notice that the products A B A B and B A B A are not equal.

This illustrates the fact that matrix multiplication is not commutative.

Is it possible for AB to be defined but not BA ?

Yes, consider a matrix A with dimension 3 × 4 3 × 4 and matrix B with dimension 4 × 2. 4 × 2. For the product AB the inner dimensions are 4 and the product is defined, but for the product BA the inner dimensions are 2 and 3 so the product is undefined.

Using Matrices in Real-World Problems

Let’s return to the problem presented at the opening of this section. We have Table 3 , representing the equipment needs of two soccer teams.

We are also given the prices of the equipment, as shown in Table 4 .

We will convert the data to matrices. Thus, the equipment need matrix is written as

The cost matrix is written as

We perform matrix multiplication to obtain costs for the equipment.

The total cost for equipment for the Wildcats is $2,520, and the total cost for equipment for the Mud Cats is $3,840.

Given a matrix operation, evaluate using a calculator.

Save each matrix as a matrix variable [ A ] , [ B ] , [ C ] , ... [ A ] , [ B ] , [ C ] , ...
Enter the operation into the calculator, calling up each matrix variable as needed.
If the operation is defined, the calculator will present the solution matrix; if the operation is undefined, it will display an error message.

Using a Calculator to Perform Matrix Operations

Find A B − C A B − C given

On the matrix page of the calculator, we enter matrix A A above as the matrix variable [ A ] , [ A ] , matrix B B above as the matrix variable [ B ] , [ B ] , and matrix C C above as the matrix variable [ C ] . [ C ] .

On the home screen of the calculator, we type in the problem and call up each matrix variable as needed.

The calculator gives us the following matrix.

Access these online resources for additional instruction and practice with matrices and matrix operations.

Dimensions of a Matrix
Matrix Addition and Subtraction
Matrix Operations
Matrix Multiplication

7.5 Section Exercises

Can we add any two matrices together? If so, explain why; if not, explain why not and give an example of two matrices that cannot be added together.

Can we multiply any column matrix by any row matrix? Explain why or why not.

Can both the products A B A B and B A B A be defined? If so, explain how; if not, explain why.

Can any two matrices of the same size be multiplied? If so, explain why, and if not, explain why not and give an example of two matrices of the same size that cannot be multiplied together.

Does matrix multiplication commute? That is, does A B = B A ? A B = B A ? If so, prove why it does. If not, explain why it does not.

For the following exercises, use the matrices below and perform the matrix addition or subtraction. Indicate if the operation is undefined.

A + B A + B

C + D C + D

A + C A + C

B − E B − E

C + F C + F

D − B D − B

For the following exercises, use the matrices below to perform scalar multiplication.

1 2 C 1 2 C

100 D 100 D

For the following exercises, use the matrices below to perform matrix multiplication.

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed.

A + B − C A + B − C

4 A + 5 D 4 A + 5 D

2 C + B 2 C + B

3 D + 4 E 3 D + 4 E

C −0.5 D C −0.5 D

100 D −10 E 100 D −10 E

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: A 2 = A ⋅ A A 2 = A ⋅ A )

B 2 A 2 B 2 A 2

A 2 B 2 A 2 B 2

( A B ) 2 ( A B ) 2

( B A ) 2 ( B A ) 2

( A B ) C ( A B ) C

A ( B C ) A ( B C )

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. Use a calculator to verify your solution.

A B C A B C

For the following exercises, use the matrix below to perform the indicated operation on the given matrix.

Using the above questions, find a formula for B n . B n . Test the formula for B 201 B 201 and B 202 , B 202 , using a calculator.

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Matrices is a plural form of a matrix, which is a rectangular array or a table where numbers or elements are arranged in rows and columns. They can have any number of columns and rows. Different operations can be performed on matrices such as addition, scalar multiplication, multiplication, transposition, etc.

There are certain rules to be followed while performing these matrix operations like they can be added or subtracted if only they have the same number of rows and columns whereas they can be multiplied if only columns in first and rows in second are exactly the same. Let us understand the different types of matrices and these rules in detail.

What are Matrices?

Matrices , the plural form of a matrix, are the arrangements of numbers, variables, symbols, or expressions in a rectangular table that contains various numbers of rows and columns. They are rectangular-shaped arrays, for which different operations like addition, multiplication, and transposition are defined. The numbers or entries in the matrix are known as its elements. Horizontal entries of matrices are called rows and vertical entries are known as columns.

Matrix Definition

A matrix is a rectangular array of numbers, variables, symbols, or expressions that are defined for the operations like subtraction, addition, and multiplications. The size of a matrix (which is known as the order of the matrix ) is determined by the number of rows and columns in the matrix. The order of a matrix with 6 rows and 4 columns is represented as a 6 × 4 and is read as 6 by 4. For example, the given matrix B is a 3 × 4 matrix and is written as $[{B}]_{3 \times 4}$:

$B = \left[\begin{array}{ccc} 2 & -1 & 3 & 5 \\ 0 & 5 & 2 & 7\\ 1 & -1 & -2 & 9 \end{array}\right]$

$Matrix definition$

Notation of Matrices

If a matrix has m rows and n columns, then it will have m × n elements . A matrix is represented by the uppercase letter, in this case, 'A', and the elements in the matrix are represented by the lower case letter and two subscripts representing the position of the element in the number of row and column in the same order, in this case, '$a_{ij}$', where i is the number of rows, and j is the number of columns. For example, in the given matrix A, element in the 3rd row and 2nd column would be $a_{32}$, can be verified in the matrix given below:

$A = \left[\begin{array}{ccc} a_{11} & a_{12} & a_{13} .. .& a_{1n} \\ a_{21} & a_{22} & a_{23} ... & a_{2n} \\ a_{31} & a_{32} & a_{33} ...& a_{3n} \\ : & : & : & : \\ a_{m1} & a_{m2} & a_{m3} ...& a_{mn} \end{array}\right] $

Calculate Matrices

We can solve matrices by performing operations on them like addition, subtraction, multiplication, and so on. Calculating matrices depends upon the number of rows and columns. For addition and subtraction, the number of rows and columns must be the same whereas, for multiplication, number of columns in the first and the number of rows in the second matrix must be equal. The basic operations that can be performed on matrices are:

Addition of Matrices

Subtraction of matrices, scalar multiplication, multiplication of matrices.

The addition of matrices can only be possible if the number of rows and columns of both the matrices are the same. While adding 2 matrices, we add the corresponding elments. i.e., (A + B) = [a$_{ij}$] + [b$_{ij}$] = [a$_{ij}$ + b$_{ij}$], where i and j are the number of rows and columns respectively. For example: $\begin{bmatrix} 2 & {-1}\\ \\ 0 & 5\end{bmatrix} + \begin{bmatrix} 0 & 2 \\ \\ 1 & -2 \end{bmatrix}\\ = \begin{bmatrix} 2+0 & {-1} +2 \\ \\ 0+1 & 5+(-2) \end{bmatrix}\\ = \begin{bmatrix} 2 & 1 \\ \\1 & 3 \end{bmatrix} $

Matrices subtraction is also possible only if the number of rows and columns of both the matrices are the same. While subtracting 2 matrices, we subtract the corresponding elements. i.e., (A - B) = [a$_{ij}$] - [b$_{ij}$] = [a$_{ij}$ - b$_{ij}$], where i and j are the row number and column number respectively. For example: $ \begin{bmatrix} 2 & {-1}\\ \\ 0 & 5 \end{bmatrix} -\begin{bmatrix} 0 & 2 \\ \\1 & -2 \end{bmatrix} \\ = \begin{bmatrix} 2-0 & {-1} -2\\ \\ 0-1 & 5-(-2) \end{bmatrix} \\ = \begin{bmatrix} 2 & -3\\ \\ -1 & 7 \end{bmatrix} $

The product of a matrix A with any number 'c' is obtained by multiplying every entry of the matrix A by c, is called scalar multiplication . i.e., (cA)$_{ij}$ = c(A$_{ij}$)

Properties of scalar multiplication in matrices

The different properties of matrices for scalar multiplication of any scalars K and l, with matrices A and B are given as,

K(A + B) = KA + KB
(K + l)A = KA + lA
(Kl)A = K(lA) = l(KA)
(-K)A = -(KA) = K(-A)

Matrices multiplication is defined only if the number of columns in the first matrix and rows in the second matrix are equal. To understand how matrices are multiplied, let us first consider a row vector $R=\left[ {{r}_{1}}\ {{r}_{2}}...{{r}_{n}} \right]$ and a column vector $C=\left[ \begin{align} \; \ {{c}_{1}} \;\\ \; \ {{c}_{2}} \; \\ \; \ \ \vdots \; \ \\ \; \ {{c}_{n}} \;\ \\ \end{align} \right]$. Then the product of R and C can be defined as

$RC=\left[ {{r}_{1}}\ \ {{r}_{2}}\ \ ...\ {{r}_{n}} \right]\ \left[ \begin{align} & \ {{c}_{1}} \\ & \ {{c}_{2}} \\ & \ \ \vdots \ \\ & \ {{c}_{n}}\ \\ \end{align} \right]\ \\ =[{{r}_{1}}{{c}_{1}}+{{r}_{2}}{{c}_{2}}+...+{{r}_{n}}{{c}_{n}}]$. For example,

$\left[ 1\ \ 3\ \ 2 \right]\ \ \left[ \begin{align} & \ \ 2 \\ & -1 \\ & \ \ 4 \\ \end{align} \right]=[7]$

Now, we will discuss matrix multiplication. It will soon become evident that to multiply 2 matrices A and B and to find AB , the number of columns in A should equal the number of rows in B .

Let A be of order m × n and B be of order n × p . The matrix AB will be of order m × p and will be obtained by multiplying each row vector of A successively with column vectors in B . Let us understand this using a concrete example:$A=\left[ \begin{matrix} {{a}_{1}} & {{a}_{2}} & {{a}_{3}} \\ {{b}_{1}} & {{b}_{2}} & {{b}_{3}} \\ {{c}_{1}} & {{c}_{2}} & {{c}_{3}} \\\end{matrix} \right]B=\left[ \begin{matrix} {{\alpha }_{1}} & {{\beta }_{1}} \\ {{\alpha }_{1}} & {{\beta }_{2}} \\ {{\alpha }_{3}} & {{\beta }_{3}} \\\end{matrix} \right]$

To obtain the element $a_{11}$ of AB, we multiply $R_1$ of A with $C_1$ of B :

To obtain the element $a_{12}$ of AB, we multiply $R_1$ of A with $C_2$ of B:

To obtain the element ${{a}_{21}}$ of AB, we multiply $R_2$ of A with $C_1$ of B:

Proceeding this way, we obtain all the elements of AB.

Let us generalize this: if A is or order m × n, and B of order n × p, then to obtain the element $ a_{ij}$ in AB, we multiply $R_i$ in A with $C_j$ in B:

Properties of Matrix Multiplication

There are different properties associated with the multiplication of matrices. For any three matrices A, B, and C:

A(BC) = (AB)C
A(B + C) = AB + AC
(A + B)C = AC + BC
A$I_m$ = A = AI n , for identity matrices I$_m$ and I n .
A$_{m\times n}$O$_{n\times p}$ = O$_{m\times p}$, where O is a null matrix.

Transpose of Matrix

The transpose of a matrix is done when we replace the rows of a matrix to the columns and columns to the rows. Interchanging of rows and columns is known as the transpose of matrices. In the matrix given below, we have row elements as row-1: 2, -3, -4, and row-2: -1, 7, -7. On transposing, we will get the elements in column-1: 2, -3, -4, and column-2: -1, 7, -7, we can check that in the image given below:

Properties of transposition in matrices

There are various properties associated with transposition. For matrices A and B, given as,

(A T ) T = A
(A + B) T = A T + B T , A and B being of the same order.
(KA) T = KA T , K is any scalar( real or complex ).
(AB) T = B T A T , A and B being conformable for the product AB. (This is also called reversal law.)

Apart from these operations, we have several other operations on matrices like finding its trace, determinant, minors and cofactors , adjoint, inverse, etc. Let us learn each of these in detail in the upcoming sections.

Trace of a Matrix

The trace of any matrix A, Tr(A) is defined as the sum of its diagonal elements. Some properties of trace of matrices are,

tr(AB) = tr(BA)
tr(A) = tr(A T )
tr(cA) = c tr(A), for a scalar 'c'
tr(A + B) = tr(A) + tr(B)

Determinant of Matrices

The determinant of a matrix is a number defined only for square matrices. It is used in the analysis of linear equations and their solution. The determinant formula helps calculate the determinant of a matrix using the elements of the matrix. Determinant of a matrix is equal to the summation of the product of the elements of a particular row or column with their respective cofactors. Determinant of a matrix A is denoted as |A|. Let say we want to find the determinant of the matrix $A = \left[\begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right] $

Then determinant formula of matrix A:

$a_{11}(-1)^{1 + 1} \!\!\left|\begin{matrix}a_{22}\!\!\!&a_{23}\\a_{32}\!\!\!&a_{33}\end{matrix}\right| \!\!+\!\! a_{12}(-1)^{1 + 2} \!\!\left|\begin{matrix}a_{21}\!\!\!&a_{23}\\a_{31}\!\!\!&a_{33}\end{matrix}\right| \!\!+\!\! a_{13}(-1)^{1 + 3} \!\!\left|\begin{matrix}a_{21}\!\!\!&a_{22}\\a_{31}\!\!\!&a_{32}\end{matrix}\right|$

Minor of Matrix

Minor for a particular element in the matrices is defined as the determinant of the matrix that is obtained when the row and column of the matrix in which that particular element lies are deleted, and the minor of the element $a_{ij}$ is denoted as $M_{ij}$. For example, for the given matrix, minor of $ a_{12}$ of the matrix $A = \left[\begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right] $ is:

$M_{12} = \left|\begin{array}{ccc} a_{21} & a_{23} \\ \\ a_{31} & a_{33} \end{array}\right|$

Similarly, we can find all the minors of the matrix and will get a minor matrix M of the given matrix A as:

$M = \left[\begin{array}{ccc} M_{11} & M_{12} & M_{13} \\ M_{21} & M_{22} & M_{23} \\ M_{31} & M_{32} & M_{33} \end{array}\right]$

Cofactor of Matrix

Cofactor of an element in the matrix A is obtained when the minor $M_{ij}$ of the matrix is multiplied with (-1) i+j . The cofactor of a matrix is denoted as $C_{ij}$. If the minor of a matrix is $M_{ij}$, then the cofactor of the matrix would be:

$C_{ij} = (-1)^{i+j} M_{ij}$

On finding all the cofactors of the matrix, we will get a cofactor matrix C of the given matrix A:

$C = \left[\begin{array}{ccc} C_{11} & C_{12} & C_{13} \\ C_{21} & C_{22} & C_{23} \\ C_{31} & C_{32} & C_{33} \end{array}\right] $

Note: Be extra cautious about the negative sign while calculating the cofactor of the matrix.

Adjoint of Matrices

The adjoint of matrices is calculated by finding the transpose of the cofactors of the elements of the given matrices. To find the adjoint of a matrix, we have to calculate the cofactors of the elements of the matrix and then transpose the cofactor matrix to get the adjoint of the given matrix. The adjoint of matrix A is denoted by adj(A). Let us understand this with an example: We have a matrix $A = \left[\begin{array}{ccc} 2 & -1 & 3 \\ 0 & 5 & 2 \\ 1 & -1 & -2 \end{array}\right] $

Then the minor matrix M of the given matrix would be:

$M = \left[\begin{array}{ccc} -8 & -2 & -5 \\ 5 & -7 & -1 \\ -17 & 4 & 10 \end{array}\right] $

We will get the cofactor matrix C of the given matrix A as:

$C = \left[\begin{array}{ccc} -8 & 2 & -5 \\ -5 & -7 & 1 \\ -17 & -4 & 10 \end{array}\right] $

Then the transpose of the cofactor matrix will give the adjoint of the given matrix:

adj(A) = C T = $\left[\begin{array}{ccc} -8 & -5 & -17 \\ 2 & -7 & -4 \\ -5 & 1 & 10 \end{array}\right] $

Inverse of Matrices

The inverse of any matrix is denoted as the matrix raised to the power (-1), i.e. for any matrix "A", the inverse matrix is denoted as A -1 . The inverse of a square matrix, A is A -1 only when: A × A -1 = A -1 × A = I . There is a possibility that sometimes the inverse of a matrix does not exist if the determinant of the matrix is equal to zero(|A| = 0). The inverse of a matrix is shown by A -1 . Matrices inverse is calculated by using the following formula:

A -1 = (1/|A|)(Adj A)

|A| is the determinant of the matrix A and |A| ≠ 0.
Adj A is the adjoint of the given matrix A.

The inverse of a 2 × 2 matrix $A = \left[\begin{array}{ccc} a_{11} & a_{12} \\ \\ a_{21} & a_{22} \end{array}\right] $ is calculated by: A -1 = $\dfrac{1}{a_{11}a_{22} - a_{12}a_{21}}\left(\begin{matrix}a_{22}&-a_{12}\\ \\-a_{21}&a_{11}\end{matrix}\right)$

Let us find the inverse of the 3 × 3 matrix we have used in the previous section: $A = \left[\begin{array}{ccc} 2 & -1 & 3 \\ 0 & 5 & 2 \\ 1 & -1 & -2 \end{array}\right] $

Since adj(A) = $\left[\begin{array}{ccc} -8 & -5 & -17 \\ 2 & -7 & -4 \\ -5 & 1 & 10 \end{array}\right] $

And on calculating the determinant , we will get |A| = -33

Therefore, A -1 = (1/-33) × $\left[\begin{array}{ccc} -8 & -5 & -17 \\ 2 & -7 & -4 \\ -5 & 1 & 10 \end{array}\right] $

Hence, A -1 = $\left[\begin{array}{ccc} 0.24 & 0.15 & 0.51 \\ -0.06 & 0.21 & 0.12 \\ 0.15 & -0.03 & -0.39 \end{array}\right] $

Types of Matrices

There are various types of matrices based on the number of elements and the arrangement of elements in them.

Row matrix: A row matrix is a matrix having a single row is called a row matrix. Example: [1, −2, 4].

Column matrix: A column matrix is a matrix having a single column is called a column matrix. Example: [−1, 2, 5] T .

Square matrix: A matrix having equal number of rows and columns is called a square matrix . For example: $B= \left[\begin{array}{ccc} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 0 &6& 1 \end{array}\right] $

Rectangular Matrix: A matrix having unequal number of rows and columns is called a rectangular matrix . For example: $B= \left[\begin{array}{ccc} 1 & 2 & 3 \\ \\ 0 & 1 & 4 \end{array}\right] $

Diagonal matrices: A matrix with all non-diagonal elements to be zeros is known as a diagonal matrix . Example: $A = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 &0 & 3 \end{array}\right] $

Identity matrices: A diagonal matrix having all the diagonal elements equal to 1 is called an identity matrix . Example: $B= \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 &0 & 1 \end{array}\right] $

Symmetric and skew-symmetric matrices: Symmetric matrices: A square matrix D of size n×n is considered to be symmetric if and only if D T = D. For example, D = $\left[\begin{array}{lll} 2 & 3 & 6 \\ 3 & 4 & 5 \\ 6 & 5 & 9 \end{array}\right] $ is a symmetric matrix because

D T = $\left[\begin{array}{lll} 2 & 3 & 6 \\ 3 & 4 & 5 \\ 6 & 5 & 9 \end{array}\right]$ = D

Skew-symmetric matrices-A square matrix F of size n×n is considered to be skew-symmetric if and only if F T = - F.

$F = \left[\begin{array}{ccc} 0 & 3 \\ \\ -3 & 0 \end{array}\right]$ is a skew-symmetric matrix because

F T = $\left[\begin{array}{cc} 0 & -3\\ \\ 3 & 0 \end{array}\right]$
-F = $\left[\begin{array}{cc} 0 & -3\\ \\ 3 & 0 \end{array}\right]$

Invertible Matrix: Any square matrix A is called invertible matrix , if there exists another matrix B, such that, AB = BA = $I_n$, where $I_n$ is an identity matrix with n × n.

Orthogonal Matrix: Any square matrix A is orthogonal if its transpose is equal to its inverse. i.e., A T = A -1

Solving a System of Equations Using Matrices

While solving the system of equations using matrices, we have three matrices A, B, and X where A is known as the coefficient matrix, B is known as the constant matrix, and X contains all the variables of the equations which is known as a variable matrix. Matrix A is of the order m × n, while B is the column matrix of the order m × 1. The product of matrix A and matrix X results in matrix B; hence, X is a column matrix as well of the order n × 1.

The matrices are arranged as:

Let's understand how to solve a system of equations using matrices with the help of an example. We have a set of two equations as given below. The equations are:

x + y = 8 2x + 3y = 10

Arrange all the coefficients, variables, and constants in the matrix in such a way that whenever we find the product of the matrices, the result obtained must result in the equation. Then the matrix equation is, AX = B where:

$A = \begin{bmatrix} 1 & 1\\ \\ 2 & 3\\ \end{bmatrix}$

$X = \begin{bmatrix} x\\ \\ y\\ \end{bmatrix}$

$B = \begin{bmatrix} 8\\ \\ 10\\ \end{bmatrix}$

To solve the equations, we need to find matrix X. It can be found by multiplying the inverse of matrix A with B, which is given as $ X = (A^{-1})B$. To find the determinant of matrix A, we will follow the below steps:

$ |A| = \begin{vmatrix} 1 & 1\\ \\ 2 & 3\\ \end{vmatrix}$

Hence, |A| = 3 - 2 = 1 $\because$ $|A| \neq 0$, it is possible to find the inverse of matrix A.

Now, by using the formula for finding the inverse of 2x2 matrix (which is mentioned in previous sections),

$A^{-1} = \begin{bmatrix} 3 & -1\\ \\ -2 & 1\\ \end{bmatrix}$

Now to find the matrix X, we'll multiply $A^{-1}$ and B. We get,

$\begin{bmatrix} 3 & -1 \\ \\ -2 & 1 \end{bmatrix} % \begin{bmatrix} 8 \\ \\ 10 \end{bmatrix} \ = \begin{bmatrix} 14 \\ \\ -6 \end{bmatrix} $

Hence, the value of matrix X is,

$X = \begin{bmatrix} 14\\ \\ -6\\ \end{bmatrix}$

Rank of a Matrix

The rank of a matrix A is defined as the maximum number of linearly independent row(or column) vectors of the matrix. That means the rank of a matrix will always be less than or equal to the number of its rows or columns. The rank of a null matrix is zero since it has no independent row or column vectors.

Eigen Values and Eigen Vectors of Matrices

If A is any square matrix of order 'n', a matrix of A - λI can be formed, where I is a unit matrix of order n, such that the number λ, called the eigenvalue and a non-zero vector v, called the eigenvector , satisfy the equation, Av = λv. λ is an eigenvalue of an n×n-matrix A if and only if A − λI n is not invertible, which is equivalent to Det(A - λI) = 0.

Matrices Formulas

There are different formulas associated with matrix operations depending upon the type of matrix. Some of the matrices formulas are listed below:

A(adj A) = (adj A) A = | A | I n
| adj A | = | A | n-1
adj (adj A) = | A | n-2 A
| adj (adj A) | = | A | (n-1)^2
adj (AB) = (adj B) (adj A)
adj (A m ) = (adj A) m ,
adj (kA) = k n-1 (adj A) , k ∈ R
adj(I n ) = I n
A is symmetric ⇒ (adj A) is also symmetric.
A is diagonal ⇒ (adj A) is also diagonal.
A is triangular ⇒ adj A is also triangular.
A is singular ⇒| adj A | = 0
A -1 = (1/|A|) adj A
(AB) -1 = B -1 A -1

Important Notes on Matrices:

Cofactor of the matrix A is obtained when the minor $M_{ij}$ of the matrix is multiplied with (-1) i+j .
Matrices are rectangular-shaped arrays.
The inverse of matrices is calculated by using the given formula: A -1 = (1/|A|)(adj A).
The inverse of a matrix exists if and only if |A| ≠ 0.

☛ Related Topics:

Matrix Calculator
Diagonal Matrix Calculator
Transpose Matrix Calculator
Matrix Addition Calculator

Solved Examples on Matrices

Example 1: Let $A=\left[ \begin{matrix} 1 & 2\\ \\ 3 & 1 \\\end{matrix} \right],\ B=\left[ \begin{matrix} 1 & 4\\ \\ 3 & -1 \\\end{matrix} \right]$. Calculate A + B.

Solution: Here, matrix A = $\left[ \begin{matrix} 1 & 2\\ \\ 3 & 1 \\\end{matrix} \right]$ matrix B = $\left[ \begin{matrix} 1 & 4\\ \\ 3 & -1 \\\end{matrix} \right]$

Using addition of matrices property, A + B = $\left[ \begin{matrix} 1 & 2 \\ \\ 3 & 1 \\\end{matrix} \right]$ + $\left[ \begin{matrix} 1 & 4\\ \\ 3 & -1 \\\end{matrix} \right]$ = $\left[ \begin{matrix} 2 & 6\\ \\ 6 & 0 \end{matrix} \right]$

Answer: Sum of matrices A and B, A + B = $\left[ \begin{matrix} 2 & 6\\ \\ 6 & 0 \end{matrix} \right]$

Example 2: Find the inverse of a matrix A =$\left[\begin{matrix}1 & -2\\ \\2 & -3 \end{matrix}\right]$.

The given matrix is A = $\left[\begin{matrix}1 & -2\\ \\2 & -3 \end{matrix}\right]$.

Using the formula of matrix inverse: A -1 = $\dfrac{1}{a_{11}a_{22} - a_{12}a_{21}}\left[\begin{matrix}a_{22}&-a_{12}\\ \\-a_{21}&a_{11}\end{matrix}\right]$

Using the inverse of matrix formula we can calculate A -1 as follows.

A -1 = $\dfrac{1}{(1× -3) - (-2 × 2)}\left[\begin{matrix}-3&2\\ \\-2&1\end{matrix}\right]$

= $\dfrac{1}{-3 +4}\left[\begin{matrix}-3&2\\ \\-2&1\end{matrix}\right]$

= $\left[\begin{matrix}-3&2\\ \\-2&1\end{matrix}\right]$

Answer: Therefore A -1 = $\left[\begin{matrix}-3&2\\ \\-2&1\end{matrix}\right]$.

Example 3: Prove that the product of the matrices A = $\left[\begin{array}{rr}1 & 2 & -1\\ 3 & 2 & 0\\ -4 & 0 & 2\end{array}\right]$ and the identity matrix of order 3x3 is the matrix itself.

The identity matrix of order 3x3 is, I = $\left[\begin{array}{rr}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{array}\right]$.

Now, AI = $\left[\begin{array}{rr}1 & 2 & -1\\ 3 & 2 & 0\\ -4 & 0 & 2\end{array}\right]$ $\left[\begin{array}{rr}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{array}\right]$

= $\left[\begin{array}{rr}1+0+0 & 0+2+0 & 0+0-1 \\ 3+0+0 & 0+2+0 & 0+0+0 \\ -4+0+0& 0+0+0& 0+0+2\end{array}\right]$

= $\left[\begin{array}{rr}1 & 2 & -1\\ 3 & 2 & 0\\ -4 & 0 & 2\end{array}\right]$

Answer: We have proved that AI = A.

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Practice Questions on Matrices

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FAQs on Matrices

What is the meaning of matrix in math.

A matrix in is an arrangement of numbers, variables, symbols, or expressions in the rectangular table which contains various numbers of rows and columns, for which the operations like addition, multiplication, transposition, etc are defined.

How to Solve Matrices?

We can solve matrices by performing matrix operations on them like addition, subtraction, multiplication, and so on. We have to take care of the orders while solving matrices.

For the addition/subtraction of 2 matrices, their orders should be the same.
For the multiplication of matrices , the number of columns of the left side matrix should be equal to the number of rows of the right side matrix.

How to Solve Systems of Equations with Matrices?

To solve the system of equations with matrices, we will follow the steps given below.

Arrange the elements of equations in matrices and find the coefficient matrix, variable matrix, and constant matrix.
Write the equations in AX = B form.
Take the inverse of A by finding the adjoint and determinant of A.
Multiply the inverse of A to matrix B, thereby finding the value of variable matrix X.

What is 3×3 Inverse Matrix Formula?

The inverse matrix formula for a 3×3 matrix is, A -1 = adj(A)/|A|; |A| ≠ 0 where A = square matrix, adj(A) = adjoint of square matrix, and A -1 = inverse matrix of A.

What is the Special Feature Of the Determinant Formula For Matrices?

The determinant of a matrix is defined only for square matrices, and this property of the determinant formula makes it unique. Also, the determinant value can be calculated by using the elements of any row or any column.

How To Calculate the Determinant of a 2×2 Matrix Using Determinant Formula?

The determinant formula for 2x2 matrix, $A =\begin{pmatrix}a &b\\ \\c&d\end{pmatrix}$ is given by the formula |A| = ad - bc.

What is the Condition for Matrix Multiplication to be Possible?

Matrix multiplication is possible only if the matrices are compatible i.e., matrix multiplication is valid only if the number of columns of the first matrix is equal to the number of rows of the second matrix.

What Are Properties of Transposition of Matrices?

For given 2 matrices, A and B, the properties of transposition of matrices can be explained as given below,

(A + B) T = A T + B T
(kA) T = kA T , k is any scalar
(AB) T = B T A T

What is the Formula for Inverse of Matrices?

The inverse matrix formula is used to determine the inverse matrix for any given matrix. The inverse of a square matrix, A is A -1 . The inverse matrix formula can be given as, A -1 = adj(A)/|A|; |A| ≠ 0, where A is a square matrix. Also for a matrix and its inverse we have A × A -1 = A -1 × A = I.

How To Use Inverse of Matrix Formula?

The inverse matrix formula can be used following the given steps:

Step 1: Find the matrix of minors for the given matrix.
Step 2: Transform the minor matrix so obtained into the matrix of cofactors .
Step 3: Find the adjoint matrix by taking the transpose of the cofactor matrix.
Step 4: Finally divide the adjoint of a matrix by its determinant.

What are the Different Types of a Matrix?

There are different types of matrices depending upon the properties of their properties. Some of them are given as,

Row matrix and column matrix
Square matrix and a rectangular matrix
Diagonal matrix
Scalar matrix
Identity matrix
Null matrix
Upper triangular matrix and lower triangular matrix
Idempotent matrix
Symmetric and skew-symmetric matrix

What are the Properties of Scalar Multiplication in Matrices?

Given the matrices A and B (both of the same order) and scalars K and l, the different properties associated with the multiplication of matrices can be given as,

What is a Matrix Polynomial?

Given a polynomial of the form, f(x) = a 0 x n + a 1 x n-1 + a 2 x n-1 + . . . + a n-1 x + a n , and A as a square matrix of order n. Then, f(A) = a 0 A n + a 1 A n-1 + a 2 A n-2 + . . . + a n-1 A + a n A + a n is called the matrix polynomial.

What is the Echelon Form of Matrices?

A matrix A = (a$_{ij}$$_{m\times n}$ is said to be of echelon form if it is in either upper triangular or lower triangular form. To convert a matrix into the echelon form, we apply elementary row operations .

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Iterative methods based on low-rank matrix for solving the Yang–Baxter-like matrix equation

Published: 16 May 2024
Volume 43 , article number 241 , ( 2024 )

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Yudan Gan 1 &
Duanmei Zhou 1

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We propose an effective matrix iteration method and a novel zeroing neural network (NZNN) model for finding numerical commuting solutions of the (time-invariant) Yang–Baxter-like matrix equation in this paper. The proposed matrix iteration method has a second-order convergence speed, and it is proved to be stable. How to proceed with initial value selection and termination criteria are discussed. Meanwhile, two numerical experiments are adopted to illustrate the superiority of the proposed matrix iteration method in computational efficiency. The NZNN model based on Tikhonov regularization for solving the time-invariant Yang–Baxter-like matrix equation is given. Besides, numerical results are provided to substantiate the efficiency, availability and superiority of the developed NZNN model for time-invariant Yang–Baxter-like matrix equation problems.

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Acknowledgements

This work is supported by the Natural Science Foundation of Jiangxi Province (Nos. 20224BAB201013, 20224BAB202004), the 2023 Higher Education Science Research Planning Project of China Association of Higher Education (No. 23SX0405), and Research Fund of Gannan Normal University (YJG-2023-12).

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Gan, Y., Zhou, D. Iterative methods based on low-rank matrix for solving the Yang–Baxter-like matrix equation. Comp. Appl. Math. 43 , 241 (2024). https://doi.org/10.1007/s40314-024-02771-x

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Yang–Baxter-like matrix equation
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Matrices with Examples and Questions with Solutions

Examples and questions on matrices along with their solutions are presented .

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Definition of a matrix, matrix entry (or element), square matrix, identity matrix, diagonal matrix, triangular matrix, transpose of a matrix, symmetric matrix, questions on matrices: part a, questions on matrices: part b, solutions to the questions in part a, solutions to the questions in part b.

The following are examples of matrices (plural of matrix ).

Example 1 The following matrix has 3 rows and 6 columns.

The entry (or element) in a row i and column j of a matrix A (capital letter A) is denoted by the symbol $(A)_{ij} $ or $ a_{ij} $ (small letter a).

In the matrix A shown below, $a_{11} = 5 $, $a_{12} = 2 $, etc ... or $ (A)_{11} = 5 $, $ (A)_{12} = 2 $, etc ... \[ \textbf{A} = \begin{bmatrix} 5 & 2 & 7 & -3 \\ -9 & -2 & -7 & 11\\ \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ \end{bmatrix} \]

A square matrix has the number of rows equal to the number of columns.

For each matrix below, determine the order and state whether it is a square matrix. \[ a) \begin{bmatrix} -1 & 1 & 0 & 3 \\ 4 & -3 & -7 & -9\\ \end{bmatrix} \;\;\;\; b) \begin{bmatrix} -6 & 2 & 0 \\ 3 & -3 & 4 \\ -5 & -11 & 9 \end{bmatrix} \;\;\;\; \\ c) \begin{bmatrix} 1 & -2 & 5 & -2 \end{bmatrix} \;\;\;\; d) \begin{bmatrix} -2 & 0 \\ 0 & -3 \end{bmatrix} \;\;\;\; e) \begin{bmatrix} 3 \end{bmatrix} \] Solutions a) order: 2 × 4. Number of rows and columns are not equal therefore not a square matrix. b) order: 3 × 3. Number of rows and columns are equal therefore this matrix is a square matrix. c) order: 1 × 4. Number of rows and columns are not equal therefore not a square matrix. A matrix with one row is called a row matrix (or a row vector). d) order: 2 × 2. Number of rows and columns are equal therefore this is square matrix. e) order: 1 × 1. Number of rows and columns are equal therefore this matrix is a square matrix.

An identity matrix I n is an n×n square matrix with all its element in the diagonal equal to 1 and all other elements equal to zero. Example 4 The following are all identity matrices. \[I_1= \begin{bmatrix} 1 \\ \end{bmatrix} \quad , \quad I_2= \begin{bmatrix} 1 & 0\\ 0& 1 \end{bmatrix} \quad , \quad I_3= \begin{bmatrix} 1 & 0 & 0\\ 0& 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \]

A diagonal matrix is a square matrix with all its elements (entries) equal to zero except the elements in the main diagonal from top left to bottom right. \[A = \begin{bmatrix} 6 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 2 \end{bmatrix} \]

An upper triangular matrix is a square matrix with all its elements below the main diagonal equal to zero. Matrix U shown below is an example of an upper triangular matrix. A lower triangular matrix is a square matrix with all its elements above the main diagonal equal to zero. Matrix L shown below is an example of a lower triangular matrix. $U = \begin{bmatrix} 6 & 2 & -5 \\ 0 & -2 & 7 \\ 0 & 0 & 2 \end{bmatrix} \qquad L = \begin{bmatrix} 6 & 0 & 0 \\ -2 & -2 & 0 \\ 10 & 9 & 2 \end{bmatrix} $

The transpose of an m×n matrix $ A $ is denoted $ A^T $ with order n×m and defined by \[ (A^T)_{ij} = (A)_{ji} \] Matrix $ A^T $ is obtained by transposing (exchanging) the rows and columns of matrix $ A $. Example 5 \[ \begin{bmatrix} 6 & 0 \\ -2 & -2\\ 10 & 9 \end{bmatrix} ^T = \begin{bmatrix} 6 & -2 & 10 \\ 0 & -2 &9\\ \end{bmatrix} \] Transpose a matrix an even number of times and you get the original matrix: $ ((A)^T)^T = A $. Transpose matrix an odd number of times and you get the transpose matrix: $ (((A)^T)^T)^T = A^T $. The transpose of any square diagonal matrix is the matrix itself. \[ \begin{bmatrix} 3 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 6 \end{bmatrix} ^T = \begin{bmatrix} 3 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 6 \end{bmatrix} \]

A square matrix is symmetric if its elements are such that $ A_{ij} = A_{ji} $ in other words $ A $ is symmetric if $A = A^T $. Example 6 Symmetric matrices \[ \begin{bmatrix} 4 & -2 & 1 \\ -2 & 5 & 7 \\ 1 & 7 & 8 \end{bmatrix} \]

Given the matrices: \[ A = \begin{bmatrix} -1 & 23 & 10 \\ 0 & -2 & -11 \\ \end{bmatrix} ,\quad B = \begin{bmatrix} -6 & 2 & 10 \\ 3 & -3 & 4 \\ -5 & -11 & 9 \\ 1 & -1 & 9 \end{bmatrix} ,\quad C = \begin{bmatrix} -3 & 2 & 9 & -5 & 7 \end{bmatrix} \\ D = \begin{bmatrix} -2 & 6 \\ -5 & 2\\ \end{bmatrix} ,\quad E = \begin{bmatrix} 3 \end{bmatrix} ,\quad F = \begin{bmatrix} 3 \\ 5 \\ -11 \\ 7 \end{bmatrix} ,\quad G = \begin{bmatrix} -6 & -4 & 23 \\ -4 & -3 & 4 \\ 23 & 4 & 9 \\ \end{bmatrix} \] a) What is the dimension of each matrix? b) Which matrices are square? c) Which matrices are symmetric? d) Which matrix has the entry at row 3 and column 2 equal to -11? e) Which matrices has the entry at row 1 and column 3 equal to 10? f) Which are column matrices? g) Which are row matrices? h) Find $ A^T , C^T , E^T , G^T $.

1) Given the matrices: \[ A = \begin{bmatrix} 23 & 10 \\ 0 & -11 \\ \end{bmatrix} ,\quad B = \begin{bmatrix} -6 & 0 & 0 \\ -1 & -3 & 0 \\ -5 & 3 & -9 \\ \end{bmatrix} ,\quad C = \begin{bmatrix} -3 & 0\\ 0 & 2 \end{bmatrix} \\ ,\quad D = \begin{bmatrix} -7 & 3 & 2 \\ 0 & 2 & 4 \\ 0 & 0 & 9 \\ \end{bmatrix} ,\quad E = \begin{bmatrix} 12 & 0 & 0 \\ 0 & 23 & 0 \\ 0 & 0 & -19\\ \end{bmatrix} \] a) Which of the above matrices are diagonal? b) Which of the above matrices are lower triangular? c) Which of the above matrices are upper triangular?

a) A: 2 × 3, B: 4 × 3, C: 1 × 5, D: 2 × 2, E: 1 × 1, F: 4 × 1, G: 3 × 3, b) D, E and G c) E and G d) B e) A and B f) E and F g) E and C h) \[ A^T = \begin{bmatrix} -1 & 0 \\ 23 & -2 \\ 10 & -11 \end{bmatrix} ,\quad C^T = \begin{bmatrix} -3 \\ 2\\ 9\\-5\\7 \end{bmatrix} ,\quad E^T = \begin{bmatrix} 3 \end{bmatrix} ,\quad G^T = \begin{bmatrix} -6 & -4 & 23\\ -4 & -3 & 4\\ 23 & 4 & 9 \end{bmatrix} \]

Definition, Formulas, Solved Example Problems - Solved Example Problems on Applications of Matrices: Solving System of Linear Equations | 12th Mathematics : UNIT 1 : Applications of Matrices and Determinants

Chapter: 12th mathematics : unit 1 : applications of matrices and determinants, solved example problems on applications of matrices: solving system of linear equations.

Solution to a System of Linear equations

(i) Matrix Inversion Method

Example 1.22.

Solve the following system of linear equations, using matrix inversion method:

5 x + 2 y = 3, 3 x + 2 y = 5 .

Then, applying the formula X = A −1 B , we get

So the solution is ( x = −1, y = 4).

Example 1.23

Solve the following system of equations, using matrix inversion method:

2 x 1 + 3 x 2 + 3 x 3 = 5,

x 1 – 2 x 2 + x 3 = -4,

3 x 1 – x 2 – 2 x 3 = 3

The matrix form of the system is AX = B,where

So, the solution is ( x 1 = 1, x 2 = 2, x 3 = −1) .

Example 1.24

Writing the given system of equations in matrix form, we get

Hence, the solution is ( x = 3, y = - 2, z = −1).

(ii) Cramer’s Rule

Example 1.25.

Solve, by Cramer’s rule, the system of equations

x 1 − x 2 = 3, 2 x 1 + 3 x 2 + 4 x 3 = 17, x 2 + 2 x 3 = 7.

First we evaluate the determinants

So, the solution is ( x 1 = 2, x 2 = - 1, x 3 = 4).

Example 1.26

In a T20 match, Chennai Super Kings needed just 6 runs to win with 1 ball left to go in the last over. The last ball was bowled and the batsman at the crease hit it high up. The ball traversed along a path in a vertical plane and the equation of the path is y = ax 2 + bx + c with respect to a xy -coordinate system in the vertical plane and the ball traversed through the points (10,8), (20,16), (30,18) , can you conclude that Chennai Super Kings won the match?

Justify your answer. (All distances are measured in metres and the meeting point of the plane of the path with the farthest boundary line is (70, 0).)

The path y = ax 2 + bx + c passes through the points (10,8), (20,16), (40, 22) . So, we get the system of equations 100 a + 10 b + c = 8, 400 a + 20 b + c = 16,1600 a + 40 b + c = 22. To apply Cramer’s rule, we find

When x = 70, we get y = 6.

So, the ball went by 6 metres high over the boundary line and it is impossible for a fielder standing even just before theboundary line to jump and catch the ball.

Hence the ball went for a super six and the Chennai Super Kings won the match.

(iii) Gaussian Elimination Method

Example 1.27.

Solve the following system of linear equations, by Gaussian elimination method :

4 x + 3 y + 6 z = 25, x + 5 y + 7 z = 13, 2 x + 9 y + z = 1.

Transforming the augmented matrix to echelon form, we get

The equivalent system is written by using the echelon form:

x + 5y + 7z = 13 , … (1)

17y + 22z = 27 , … (2)

199z = 398 . … (3)

Substituting z = 2, y = -1 in (1), we get x = 13 - 5 × (−1 ) − 7 × 2 = 4 .

So, the solution is ( x =4, y = - 1, z = 2 ).

Note. The above method of going from the last equation to the first equation is called the method of back substitution .

Example 1.28

The upward speed v ( t ) of a rocket at time t is approximated by v(t) = at 2 + bt + c, 0 ≤ t ≤ 100 where a, b, and c are constants. It has been found that the speed at times t = 3, t = 6 , and t = 9 seconds are respectively, 64, 133, and 208 miles per second respectively. Find the speed at time t = 15 seconds. (Use Gaussian elimination method.)

Since v (3) =64, v (6) = 133 and v (9) = 208 , we get the following system of linear equations

9a +3b + c = 64 ,

36a + 6b + c = 133,

81a + 9b + c = 208 .

We solve the above system of linear equations by Gaussian elimination method.

Reducing the augmented matrix to an equivalent row-echelon form by using elementary row operations, we get

Writing the equivalent equations from the row-echelon matrix, we get

9a + 3b + c = 64, 2b + c = 41, c= 1.

By back substitution, we get

So, we get v (t) = 1/3 t 2 + 20t + 1.

Hence, v (15) = 1/3 (225) + 20(15) + 1 = 75 + 300 + 1 = 376

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A matrix over a field is a function from to , where and are the sets and .A matrix is usually represented as a rectangular array of scalars from the field, such that each column belongs to the vector space, where is the number of rows. If a matrix has rows and columns, its order is said to be , and it is written as .. The element in the row and column of is written as .
7.6: Matrices and Matrix Operations
Solution. First, we check the dimensions of the matrices. Matrix \ (A\) has dimensions \ (2 × 2\) and matrix \ (B\) has dimensions \ (2 × 2\). The inner dimensions are the same so we can perform the multiplication. The product will have the dimensions \ (2 × 2\). We perform the operations outlined previously.
9.6: Solving Systems with Gaussian Elimination
R1 ↔ R2 → [1 − 1 12 2 3 6] We now have a 1 as the first entry in row 1, column 1. Now let's obtain a 0 in row 2, column 1. This can be accomplished by multiplying row 1 by − 2, and then adding the result to row 2. − 2R1 + R2 = R2 → [1 − 1 12 0 5 5] We only have one more step, to multiply row 2 by 1 5.
How to master the seven-step problem-solving process
In this episode of the McKinsey Podcast, Simon London speaks with Charles Conn, CEO of venture-capital firm Oxford Sciences Innovation, and McKinsey senior partner Hugo Sarrazin about the complexities of different problem-solving strategies.. Podcast transcript. Simon London: Hello, and welcome to this episode of the McKinsey Podcast, with me, Simon London.
4.6: Solve Systems of Equations Using Matrices
If you missed this problem, review . Solve: $0.25p+0.25(x+4)=5.20$. ... Write the Augmented Matrix for a System of Equations. Solving a system of equations can be a tedious operation where a simple mistake can wreak havoc on finding the solution. An alternative method which uses the basic procedures of elimination but with notation that is ...
Matrices
This topic covers: - Adding & subtracting matrices - Multiplying matrices by scalars - Multiplying matrices - Representing & solving linear systems with matrices - Matrix inverses - Matrix determinants - Matrices as transformations - Matrices applications
How to Solve Matrices (with Pictures)
It will also be easier, for most steps in solving the matrix, to be able to write your fractions in improper form, and then convert them back to mixed numbers for the final solution. Therefore, the number 1 2/3 is easier to work with if you write it as 5/3. For example, the first row (R1) of our sample problem begins with the terms [3,1,-1,9].
7.5 Matrices and Matrix Operations
To solve a problem like the one described for the soccer teams, we can use a matrix, which is a rectangular array of numbers. A row in a matrix is a set of numbers that are aligned horizontally. A column in a matrix is a set of numbers that are aligned vertically. Each number is an entry, sometimes called an element, of the matrix. Matrices ...
Matrix word problem: prices (video)
Toilet paper in Duluth, Minnesota cost 3.99 a package while toilet paper in New York City cost 8.95 a package. In Duluth, toothpaste costs $1.95 a tube while in New York City it costs $5.25 a tube. The data for this can be encoded in the following grocery matrix. Let's see if this makes sense.
Matrices
A matrix is a rectangular array of numbers, variables, symbols, or expressions that are defined for the operations like subtraction, addition, and multiplications. The size of a matrix (which is known as the order of the matrix) is determined by the number of rows and columns in the matrix.The order of a matrix with 6 rows and 4 columns is represented as a 6 × 4 and is read as 6 by 4.
Matrix Calculator
To multiply two matrices together the inner dimensions of the matrices shoud match. For example, given two matrices A and B, where A is a m x p matrix and B is a p x n matrix, you can multiply them together to get a new m x n matrix C, where each element of C is the dot product of a row in A and a column in B.
Iterative methods based on low-rank matrix for solving the ...
In fact, the Yang-Baxter-like matrix equation has infinite solutions for an arbitrary matrix A.Finding all the solutions that satisfy the matrix equations, which is equivalent to solving a system of $n^2$ unknowns with $n^2$ quadratic equations, is indeed a challenging task.. For a particular A, the process of solving matrix equations often involves direct computation using the Jordan ...
Matrices and Determinants: Problems with Solutions
Define the term identity matrix (unit matrix). All elements are ones. Ones on one of the diagonals and zeros elsewhere. Ones on the first row and column and zeros elsewhere. Ones on the main diagonal and zeros elsewhere. Problem 5. Write the following system of equations as an augmented matrix.
Matrices with Examples and Questions with Solutions
The following are examples of matrices (plural of matrix). An m × n (read 'm by n') matrix is an arrangement of numbers (or algebraic expressions ) in m rows and n columns. Each number in a given matrix is called an element or entry. A zero matrix has all its elements equal to zero. Example 1 The following matrix has 3 rows and 6 columns.
Solved Example Problems on Applications of Matrices: Solving System of
Example 1.27. Solve the following system of linear equations, by Gaussian elimination method : 4x + 3y + 6z = 25, x + 5 y + 7z = 13, 2x + 9 y + z = 1.. Solution. Transforming the augmented matrix to echelon form, we get
A Nonlinear Programming Approach to Solving Interval-Valued
To solve these matrix games, we formulate two interval-valued intuitionistic hesitant fuzzy programming problems. Preserving the hesitant nature of the payoffs to determine the optimal strategies, these two problems are transformed into two nonlinear programming problems. This transformation involves using mathematical operations for IIHFEs.