matrices and row operations assignment quizlet

Row Operations of Matrices

Description of matrices and elementary row operations

There are three types of elementary matrix row operations, corresponding to the operations that apply to equations to eliminate variable

Adding a multiple of one row to another row

Multiplying of a row by a non-zero scalar

Interchange of two rows

These operations can be done manually, but also by matrices multiplication with a given matrix and some modified identity matrix. See the three examples below.

Adding a multiple of one row to another

Placing \(k\) in the second column of row 3 of the identity matrix

then multiplying the matrices.

This has k-times the values of corresponding elements of row 2 added to those of row 3 of the matrix.

The value of the determinant in the result is identical to the value of the source matrix \(A\)

Multiplying a row by a non-zero scalar:

The value of the determinant in the result is \(k\) times the value of the source matrix \(A\)

Interchanging two rows

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Study Guides > College Algebra

Performing row operations on a matrix.

Now that we can write systems of equations in augmented matrix form, we will examine the various row operations that can be performed on a matrix, such as addition, multiplication by a constant, and interchanging rows. Performing row operations on a matrix is the method we use for solving a system of equations. In order to solve the system of equations, we want to convert the matrix to row-echelon form , in which there are ones down the main diagonal from the upper left corner to the lower right corner, and zeros in every position below the main diagonal as shown.

• In any nonzero row, the first nonzero number is a 1. It is called a leading 1.
• Any all-zero rows are placed at the bottom on the matrix.
• Any leading 1 is below and to the right of a previous leading 1.
• Any column containing a leading 1 has zeros in all other positions in the column.
• Interchange rows. (Notation: [latex]{R}_{i}\leftrightarrow {R}_{j}[/latex] )
• Multiply a row by a constant. (Notation: [latex]c{R}_{i}[/latex] )
• Add the product of a row multiplied by a constant to another row. (Notation: [latex]{R}_{i}+c{R}_{j}[/latex])

A General Note: Gaussian Elimination

How to: given an augmented matrix, perform row operations to achieve row-echelon form..

• The first equation should have a leading coefficient of 1. Interchange rows or multiply by a constant, if necessary.
• Use row operations to obtain zeros down the first column below the first entry of 1.
• Use row operations to obtain a 1 in row 2, column 2.
• Use row operations to obtain zeros down column 2, below the entry of 1.
• Use row operations to obtain a 1 in row 3, column 3.
• Continue this process for all rows until there is a 1 in every entry down the main diagonal and there are only zeros below.
• If any rows contain all zeros, place them at the bottom.

Example 2: Solving a [latex]2\times 2[/latex] System by Gaussian Elimination

Example 3: using gaussian elimination to solve a system of equations, example 4: solving a dependent system, example 5: performing row operations on a 3×3 augmented matrix to obtain row-echelon form, licenses & attributions, cc licensed content, specific attribution.

• Precalculus. Provided by: OpenStax Authored by: OpenStax College. Located at: https://cnx.org/contents/ [email protected] :1/Preface. License: CC BY: Attribution .

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• 5. Multiple Choice Edit 30 seconds 1 pt What comes first in the order? Row Column
• 11. Multiple Choice Edit 30 seconds 1 pt How many rows are in a 7 x 3 matrix? 7 3 21 10
• 12. Multiple Choice Edit 30 seconds 1 pt How many columns are in a 5 x 4 matrix? 5 4 20 9
• 20. Multiple Choice Edit 15 minutes 1 pt What must be true in order to ADD two matrices? They must be square. The dimensions must be equal. The determinant can't equal 0. The column of the 1st must equal the row of the 2nd.
• 25. Multiple Choice Edit 45 seconds 1 pt Given matrices A & B:  A is a (4x33) matrix B is a (33x1) matrix  What are the dimensions of the product of A times B ? 4x1 4x33 33x1 Matrix does not exist

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not possible because rows do not match columns

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Basic Matrix Row Operations Tips

There are four basic operations in mathematics: adding, subtracting, dividing and multiplying. It looks easy for a mathematician or a student who has maths as his or her major. In fact, these operations are quite simple: 2+2=4, 5-4=1 and so on. However, mathematics would have never become so much popular if it was so easy. An math homework may take ages for an ordinary high school, college or university student to find correct solutions. That is why many students require homework help in mathematics. There is nothing wrong in having problems or questions in maths. After all, we are not all geniuses. At the same time, it is still possible to get correct answers taking advantage of tips and special custom writing service companies which can be found online.

As already said above, there are 4 basic operations in maths. In case of matrices there are only three raw operations that this article will focus on.

Row switching. Every matrix consists of rows and columns of numbers. As the name of the operation suggests we switch rows of numbers and indicate substitutions with arrows. A word of advice – make sure you correctly copy the rows when switching them.

Row multiplication. In such a case you can either multiply rows of the two matrices or multiply a matrix by a certain number (in fact, anything you like). Thus, in case of two matrices you should multiply relevant numbers of the rows (for example, first right top numbers of matrix A and matrix B and so on). In case you multiply the whole matrix by one number, you perform multiplication operation with all matrix numbers.

Row addition. This operation is quite easy. One has to add relevant number of matrices (number 1 row 1 matrix A plus number 1 row 1 matrix B, and so on).

Students who need maths help can look for an online maths solver which a service that solves maths equations. In such a way you will not only get solutions to maths tasks but also learn solving “mechanics.”

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Answers to Online Quizlet 1.1

Question 1 . True or False: Every elementary row operation is reversible.

Answer : True

Question 2. A 5 x 6 matrix has six rows.

Answer: False. It has 5 rows and 6 columns.

Question 3. Elementary row operations on an augmented matrix never change the solution set of the associated linear system

Answer : True. Because of the answer to question 1.

Return to Math 308A Assignment Page.

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Matrix Row Operations

Row Operations Worked Example

What are "operations" in math?

"Operations" is mathematician-ese for "procedures" that you can do with things. For instance, the four basic operations on numbers are addition, subtraction, multiplication, and division.

Content Continues Below

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What are the three matrix row operations?

For matrices, there are three basic row operations; that is, there are three procedures that you can do with the rows of a matrix. These operations are:

• Row swapping: You pick two rows of a matrix, and switch them for each other. For instance, you might take the third row and move it to the fifth row, and put the fifth row where the third had been.
• Row multiplication: You can multiply any row by any non-zero value. For instance, if the entries in one row are all even, you might multiply the row by ½, making the entries smaller and thus easier to work with.
• Row addition: You can take the entries of one row, and add them to the entries of another row. For instance, if one row had a leading entry of −3 and another row has a leading entry of 4 , you could add the former row to the latter, giving yourself a leading 1 where there had been a 4 .

What does row-swapping look like?

Row-swapping involves nothing more than taking two rows and switching their positions. For instance, given the following matrix:

...you can switch the rows around to put the matrix into a nicer row arrangement like this:

To arrive at the above matrix, the first row of the original matrix was moved to the third row of the new matrix, the second row was moved to the first row, and the third was moved to the second row.

Row-switching is often indicated by drawing arrows, like this:

When switching rows around, be careful to copy the entries correctly.

What does row multiplication look like?

Row multiplication is nothing more than picking a row of the matrix, and multiplying it by some number. For instance, given the following matrix:

...you can multiply the first row by −1 to get a positive leading value in the first row:

This row multiplication is often indicated by using an arrow with multiplication listed on top of it, like this:

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The −1 R 1 indicates the actual operation that was executed to get from the original matrix to the new one. The −1 says that we multiplied by a negative 1 ; the R 1 says that we were working with the first row.

Note that the second and third rows were copied down, unchanged, into the new matrix. The multiplication only applied to the first row, so the entries for the other two rows were just carried along unchanged.

You can multiply by anything you like. For instance, to get a leading 1 in the third row of the previous matrix, you can multiply the third row by a negative one-half:

Since you weren't doing anything with the first and second rows, those entries were just copied over unchanged into the new matrix.

You can do more than one row multiplication within the same step, so you could have done the two above steps in just one step, like this:

It is a good idea to use some form of notation (such as the arrows and subscripts above) so you can keep track of your work. Matrices are very messy, especially if you're doing them by hand, and notes can make it easier to check your work later. It'll also impress your teacher.

What does row addition look like?

Row addition is nothing more than taking the entries of one row and adding them to the entries of another row. Row addition is similar to the addition method for solving systems of linear equations. For instance, suppose you have the following system of equations:

x + 3 y = 1 − x + y = 3

You could start solving this system by adding down the columns to get 4 y  = 4 :

Then you'd have solved for the value of y , and back-solved for the value of x .

You can do something similar with matrices. For instance, given the following matrix:

...you can reduce (that is, you can get more leading zeroes in) the second row by adding the first row to it. (The general goal with matrices at this stage being to get a leading 1 — or else leading 0 's followed by a 1 — at the beginning of each matrix row.)

When you were reducing the two-equation linear system by adding, you drew an "equals" bar across the bottom of the second equation and added down. When you are using addition on a matrix, you'll need to grab some scratch paper, because you don't want to try to do the work inside the matrix. So add the two rows on your scratch paper:

Scratch work — don't hand this in!

The result of the addition is your new second row; you will write it in place of the old second row. The result will look like this:

In the above, the notation R 1  +  R 2 " on the arrow means "I added row one to row two, and this is the result I got, being a new row two". Since row one didn't actually change, and since we didn't do anything with row three, these rows get copied into the new matrix unchanged.

Note: You can work with a row or work on , but not both. You cannot, for instance, multiply the second row by 5 , and then add the result to the second row. You can, however, add the multiplied second-row entries to any of the other rows. This means that you're working with the second row, but on one or another of the other rows. With or on, but not both!

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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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• Authors: Jay Abramson
• Publisher/website: OpenStax
• Book title: College Algebra 2e
• Publication date: Dec 21, 2021
• Location: Houston, Texas
• Book URL: https://openstax.org/books/college-algebra-2e/pages/1-introduction-to-prerequisites
• Section URL: https://openstax.org/books/college-algebra-2e/pages/7-5-matrices-and-matrix-operations

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Elementary Row Operations

There are many applications of elementary row operations. They can be used to solve a system of equations in an easy way, they can be used to find the rank of a matrix etc. The elementary row transformations are also used to find the inverse of a matrix A without using any formula like A -1 = (adj A) / (det A).

Let us see how to apply inverse row operations for performing multiple things in an easy way.

What are Elementary Row Operations?

While applying the elementary row operations, we usually represent the first row by R₁, the second row by R₂, and so on. There are primarily three types of elementary row operations:

• Interchanging two rows. For example, interchanging the first and second rows is shown by R₁ ↔ R₂.
• Multiplying/dividing a row by a scalar. For example, if the first row (all elements of the first row) is multiplied by some scalar, say 3, it is shown as R₁ → 3R₁.
• Multiplying/dividing a row by some scalar and adding/subtracting to the corresponding elements of another row. For example, if the first row is multiplied by 3, and added to the second row, we can write it either as R₁ → 3R₁ + R₂ (or) R₂ → R₂ + 3R₁.

It is a common practice to write the same row on the left side of the arrow and on the very first occurrence of the right side of the arrow.

Elementary Row Operations to Solve a System of Equations

We can solve a system of equations written in matrix form AX = B, by writing the augmented matrix [A B] and applying the elementary row operations on it to convert it into the echelon form (preferably the upper triangular form). Applying all the above three row operations do not alter the augmented matrix as:

• Interchanging two rows is nothing but swapping two equations of the system and this doesn't affect the solution.
• Multiplying a row by some scalar does not alter the augmented matrix, as we always can multiply both sides of an equation by a scalar without affecting the equation.
• Multiplying one row by scalar and adding it to the other row is nothing but multiplying an equation by a scalar and adding it to another equation and we usually do this to solve a system of equations.

This process of applying row operations to solve a system is known as Gauss elimination. We can see an example of applying row transformations to solve a system of equations in the " Elementary Row Operations Examples " section below.

Elementary Row Operations to Find Inverse of a Matrix

To find the inverse of a square matrix A, we usually apply the formula, A -1 = (adj A) / (det A). But this process is lengthy as it involves many steps like calculating cofactor matrix, adjoint matrix, determinant, etc. To make this process easy, we can apply the elementary row operations. Here are the steps for doing the same.

• Consider the augmented matrix [A | I], where I is the identity matrix that is of the same order as A.
• Apply row transformations to convert the left side matrix A into I.
• Then the right side matrix (that replaces the original matrix I) is nothing but A -1 .

To see how to find the inverse of a 2x2 matrix and the inverse of a 3x3 matrix using elementary row operations, click on the respective links.

Elementary Row Operations to Find Determinant

Usually, we find the determinant of a matrix by finding the sum of the products of the elements of a row or a column and their corresponding cofactors. But this process is difficult if the terms of the matrix are expressions. But we can apply the elementary row operations to find the determinant easily. But some of the row operations affect the determinant in the following ways:

• Interchanging two rows of a determinant changes its sign.
• Multiplying a row by some scalar multiplies the determinant by the same scalar.
• Multiplying a row by some scalar and adding the result to another row doesn't alter the determinant.

To see how to find the determinant of a matrix by elementary row operations, click here .

Elementary Row Operations to Find Rank of a Matrix

The rank of a matrix is the number of linearly independent rows (or columns) in it. We can apply the elementary row operations on the matrix to find its rank in two ways:

• We can convert it into Echelon form and count the number of non-zero rows in it which gives its rank.
• We can convert it int the normal form \(\left[\begin{array}{ll} lᵣ & 0 \\ \\ 0 & 0 \end{array}\right]\), where Iᵣ is the identity matrix of order r. Then the rank of the matrix = r.

To understand these two methods with examples, click here .

☛ Related Topics:

• System of Equations Calculator
• Determinant Calculator
• Inverse Matrix Calculator
• Matrix Calculator

Elementary Row Operations Examples

Example 1: Perform the following elementary row operations on the matrix A = \(\left[\begin{array}{rrr} 1 & 2 & -1 \\ 3 & 2 & 0 \\ -4 & 0 & 2 \end{array}\right]\): (a) R₁ ↔ R₂ (b) R₂ → R₂ - 5R₁.

(a) R₁ ↔ R₂ means swapping (or interchanging) the first two rows.

Then the result is \(\left[\begin{array}{rrr} 3 & 2 & 0 \\ 1 & 2 & -1 \\ -4 & 0 & 2 \end{array}\right]\).

(b) We have R₁ (the first row) = [1 2 -1].

Then -5R₁ = [-5 -10 5].

R₂ - 5R₁ = [3 2 0] + [-5 -10 5] = [-2 -8 5]

R₂ → R₂ - 5R₁ means replace R₂ by the row obained by doing R₂ - 5R₁. Then the resultant matrix is \(\left[\begin{array}{rrr} 1 & 2 & -1 \\ -2 & -8 & 5 \\ -4 & 0 & 2 \end{array}\right]\).

Answer: (a) \(\left[\begin{array}{rrr} 3 & 2 & 0 \\ 1 & 2 & -1 \\ -4 & 0 & 2 \end{array}\right]\) (b) \(\left[\begin{array}{rrr} 1 & 2 & -1 \\ -2 & -8 & 5 \\ -4 & 0 & 2 \end{array}\right]\).

Example 2: Solve the following system of equations using elementary row transformations: 2x - y + 3z = 8, -x + 2y + z = 4, and 3x + y - 4z = 0.

The matrix equation of the given system is:

\(\left[\begin{array}{ccc} 2 & -1 & 3 \\ -1 & 2 & 1 \\ 3 & 1 & -4 \end{array}\right]\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{l} 8 \\ 4 \\ 0 \end{array}\right]\)

The augmented matrix is,

[A B] = \(\left[\begin{array}{ccc:c} 2 & -1 & 3 & 8 \\ -1 & 2 & 1 & 4 \\ 3 & 1 & -4 & 0 \end{array}\right]\)

Here we convert the last two elements of the first column (-1 and 3) to be zero. We use R₁ in this process.

Apply R₂ → 2R₂ + R₁ and R₃ → 2R₃ - 3R₁, we get:

= \(\left[\begin{array}{ccc:c} 2 & -1 & 3 & 8 \\ 0 & 3 & 5 & 16 \\ 0 & 5 & -17 & -24 \end{array}\right]\)

We convert the last element of the second column (5) to be a zero. We use R₂ in this process.

Now, apply R₃ → 3R₃ - 5R₂,

= \(\left[\begin{array}{ccc:c} 2 & -1 & 3 & 8 \\ 0 & 3 & 5 & 16 \\ 0 & 0 & -76 & -152 \end{array}\right]\)

Now, we expand the above matrix as equations:

2x - y + 3z = 8 ... (1)

3y + 5z = 16 ... (2)

-76z = -152 ... (3)

From (3), z = (-152) / (-76) = 2.

From (2), 3y + 5(2) = 16 ⇒ 3y = 6 ⇒ y = 2.

From (1), 2x - 2 + 3 (2) = 8 ⇒ 2x = 4 ⇒ x = 2.

Answer: (x, y, z) = (2, 2, 2).

Example 3: Find the inverse of the matrix A = \(\left[\begin{array}{ccc} -2 & 1 & 3 \\ 0 & -1 & 1 \\ 1 & 2 & 0 \end{array}\right]\) using elementary row operations.

Consider the augmented matrix formed by A and the identity matrix I.

[A | I] = \(\left[\begin{array}{ccc:ccc} 1 & 0 & 0 & -2 & 1 & 3 \\ 0 & 1 & 0 & 0 & -1 & 1 \\ 0 & 0 & 1 & 1 & 2 & 0 \end{array}\right]\)

We will convert the right side matrix as the identity matrix.

Apply R₃ → 2R₃ + R₁,

= \(\left[\begin{array}{ccc:ccc} 1 & 0 & 0 & -2 & 1 & 3 \\ 0 & 1 & 0 & 0 & -1 & 1 \\ 1 & 0 & 2 & 0 & 5 & 3 \end{array}\right]\)

Now apply R₁ → R₁ + R₂ and R₃ → R₃ + 5R₂,

\(\left[\begin{array}{ccc:ccc} 1 & 1 & 0 & -2 & 0 & 4 \\ 0 & 1 & 0 & 0 & -1 & 1 \\ 1 & 5 & 2 & 0 & 0 & 8 \end{array}\right]\)

Apply R₁ → 2R₁ - R₃ and R₂ → 8R₂ - R₃,

\(\left[\begin{array}{ccc:ccc} 1 & -3 & -2 & -4 & 0 & 0 \\ -1 & 3 & -2 & 0 & -8 & 0 \\ 1 & 5 & 2 & 0 & 0 & 8 \end{array}\right]\)

Now divide R₁ by -4, R₂ by -8, and R₃ by 8:

\(\left[\begin{array}{ccc:ccc} -1 / 4 & +3 / 4 & +2 / 4 & 1 &0 &0 \\ +1 / 8 & -3 / 8 & 2 / 8 & 0 &1 & 0 \\ 1 / 8 & 5 / 8 & 2 / 8 & 0 & 1 & 0 \end{array}\right] .\)

Now, the right side matrix got converted into I. Hence, the left side matrix is A -1 .

Answer: A -1 = \(\left[\begin{array}{ccc:ccc} -1 / 4 & 3 / 4 & 2 / 4 \\ 1 / 8 & -3 / 8 & 2 / 8 \\ 1 / 8 & 5 / 8 & 2 / 8 \end{array}\right] .\)

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Practice Questions on Elementary Row Operations

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FAQs on Elementary Row Operations

How to apply elementary row operations.

We can apply three types of elementary row operations :

• We can interchange two rows.
• We can multiply/divide any row(s) by a number.
• We can multiply/divide a row by some number and add/subtract it to another row.

How to Know Which Elementary Row Operations Have to be Applied to Solve a System of Equations?

While solving the system of equations (3x3) AX = B using augmented matrix [A B]:

• First aim at making the last two elements of the first column as zeros. Use row 1 for this process.
• Then aim at making the last element of the second column as zero. Use row 2 for this process.

Then the matrix gets converted into the upper triangular matrix . Expand it back into equations and solve them easily.

Do Elementary Row Operations Affect the Determinant?

Some row operations affect the determinant . Swapping two rows changes the sign of the determinant. Multiplying a row by some number multiplies the actual determinant also by the same factor. But multiplying a row by some number and adding it to the other row does not affect the determinant.

How to Apply Elementary Row Transformations to Find the Rank of a Matrix?

To find the rank of a matrix, convert it into echelon form (upper triangular matrix or lower triangular matrix) by applying elementary row operations. Finally, count the number of non-zero rows in it. This gives the rank of the matrix.

Do Elementary Row Operations Affect the System of Equations?

No, any type of elementary row operation does not affect the system of equations. Thus, applying row transformations is an effective way of solving a system of equations .

Explain the Process of Using Elementary Row Operations to Find Inverse of a Matrix.

To find the inverse of a square matrix A:

• Take the matrix [A | I], where I is the identity matrix of same order as that of A.
• Apply elementary row transformations to convert it to the form [I | a square matrix ]
• "A square matrix" in the above step gives A -1 .