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])
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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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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.
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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
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- 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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4.6E: Exercises
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Practice Makes Perfect
Write the Augmented Matrix for a System of Equations
In the following exercises, write each system of linear equations as an augmented matrix.
ⓐ \(\left\{ \begin{array} {l} 3x−y=−1\\ 2y=2x+5\end{array} \right.\) ⓑ \(\left\{ \begin{array} {l} 4x+3y=−2\\ x−2y−3z=7 \\ 2x−y+2z=−6 \end{array} \right.\)
ⓐ \(\left\{ \begin{array} {l} 2x+4y=−5\\ 3x−2y=2\end{array} \right.\) ⓑ \(\left\{ \begin{array} {l} 3x−2y−z=−2\\ −2x+y=5 \\ 5x+4y+z=−1 \end{array} \right.\)
ⓐ \(\left[ \begin{matrix} 2 &4 &−5 \\ 3 &−2 &2 \end{matrix} \right]\) ⓑ \(\left[ \begin{matrix} 3 &−2 &−1 &−2 \\ −2 &1 &0 &5 \\ 5 &4 &1 &−1 \end{matrix} \right]\)
ⓐ \(\left\{ \begin{array} {l} 3x−y=−4 \\ 2x=y+2 \end{array} \right.\) ⓑ \(\left\{ \begin{array} {l} x−3y−4z=−2 \\ 4x+2y+2z=5 \\ 2x−5y+7z=−8 \end{array} \right.\)
ⓐ \(\left\{ \begin{array} {l} 2x−5y=−3 \\ 4x=3y−1 \end{array} \right.\) ⓑ \(\left\{ \begin{array} {l} 4x+3y−2z=−3 \\ −2x+y−3z=4 \\ −x−4y+5z=−2 \end{array} \right.\)
ⓐ \(\left[ \begin{matrix} 2 &−5 &−3 \\ 4 &−3 &−1 \end{matrix} \right]\) ⓑ \(\left[ \begin{matrix} 4 &3 &−2 &−3 \\ −2 &1 &−3 &4 \\ −1 &−4 &5 &−2 \end{matrix} \right]\)
Write the system of equations that corresponds to the augmented matrix.
\(\left[ \begin{array} {cc|c} 2 &−1 &4 \\ 1 &−3 &2 \end{array} \right]\)
\(\left[ \begin{array} {cc|c} 2 &−4 &-2 \\ 3 &−3 &-1 \end{array} \right]\)
\(\left\{ \begin{array} {l} 2x−4y=−2 \\ 3x−3y=−1 \end{array} \right.\)
\(\left[ \begin{array} {ccc|c} 1 &0 &−3 &-1 \\ 1 &−2 &0 &-2 \\ 0 &−1 &2 &3 \end{array} \right]\)
\(\left[ \begin{array} {ccc|c} 2 &−2 &0 &-1 \\ 0 &2 &−1 &2 \\ 3 &0 &−1 &-2 \end{array} \right]\)
\(\left\{ \begin{array} {l} 2x−2y=−1 \\ 2y−z=2 \\ 3x−z=−2 \end{array} \right.\)
Use Row Operations on a Matrix
In the following exercises, perform the indicated operations on the augmented matrices.
\(\left[ \begin{array} {cc|c} 6 &−4 &3 \\ 3 &−2 &1 \end{array} \right]\)
ⓐ Interchange rows 1 and 2
ⓑ Multiply row 2 by 3
ⓒ Multiply row 2 by \(−2\) and add row 1 to it.
\(\left[ \begin{array} {cc|c} 4 &−6 &-3 \\ 3 &2 &1 \end{array} \right]\)
ⓑ Multiply row 1 by 4
ⓒ Multiply row 2 by 3 and add row 1 to it.
ⓐ \(\left[ \begin{matrix} 3 &2 &1 \\ 4 &−6 &−3 \end{matrix} \right]\) ⓑ \(\left[ \begin{matrix} 12 &8 &4 \\ 4 &−6 &−3 \end{matrix} \right]\) ⓒ \(\left[ \begin{matrix} 12 &8 &4 \\ 24 &−10 &−5 \end{matrix} \right]\)
\(\left[ \begin{array} {ccc|c} 4 &−12 &−8 &16 \\ 4 &−2 &−3 &-1 \\ −6 &2 &−1 &-1 \end{array} \right]\)
\(\left[ \begin{array} {ccc|c} 6 &−5 &2 &3 \\ 2 &1 &−4 &5 \\ 3 &−3 &1 &-1 \end{array} \right]\)
ⓐ \(\left[ \begin{matrix} 2 &1 &−4 &5 \\ 6 &−5 &2 &3 \\ 3 &−3 &1 &−1 \end{matrix} \right]\) ⓑ \(\left[ \begin{matrix} 2 &1 &−4 &5 \\ 6 &−5 &2 &3 \\ 3 &−3 &1 &−1 \end{matrix} \right]\) ⓒ \(\left[ \begin{matrix} 2 &1 &−4 &5 \\ 6 &−5 &2 &3 \\ −4 &7 &−6 &7 \end{matrix} \right]\)
Perform the needed row operation that will get the first entry in row 2 to be zero in the augmented matrix: \(\left[ \begin{array} {cc|c} 1 &2 &5 \\ −3 &−4 &-1 \end{array} \right]\)
Perform the needed row operations that will get the first entry in both row 2 and row 3 to be zero in the augmented matrix: \(\left[ \begin{array} {ccc|c} 1 &−2 &3 &-4 \\ 3 &−1 &−2 &5 \\ 2 &−3 &−4 &1 \end{array} \right]\)
\(\left[ \begin{matrix} 1 &−2 &3 &−4 \\ 0 &5 &−11 &17 \\ 0 &1 &−10 &7 \end{matrix} \right]\)
Solve Systems of Equations Using Matrices
In the following exercises, solve each system of equations using a matrix.
\(\left\{ \begin{array} {l} 2x+y=2 \\ x−y=−2 \end{array} \right.\)
\(\left\{ \begin{array} {l} 3x+y=2 \\ x−y=2 \end{array} \right.\)
\((1,−1)\)
\(\left\{ \begin{array} {l} −x+2y=−2 \\ x+y=−4 \end{array} \right.\)
\(\left\{ \begin{array} {l} −2x+3y=3 \\ x+3y=12 \end{array} \right.\)
\(\left\{ \begin{array} {l} 2x−3y+z=19 \\ −3x+y−2z=−1 \\ 5x+y+z=0 \end{array} \right.\)
\(\left\{ \begin{array} {l} 2x−y+3z=−3 \\ −x+2y−z=10 \\ x+y+z=5 \end{array} \right.\)
\((−2,5,2)\)
\(\left\{ \begin{array} {l} 2x−6y+z=3 \\ 3x+2y−3z=2 \\ 2x+3y−2z=3 \end{array} \right.\)
\(\left\{ \begin{array} {l} 4x−3y+z=7 \\ 2x−5y−4z=3 \\ 3x−2y−2z=−7 \end{array} \right.\)
\((−3,−5,4)\)
\(\left\{ \begin{array} {l} x+2z=0 \\ 4y+3z=−2 \\ 2x−5y=3 \end{array} \right.\)
\(\left\{ \begin{array} {l} 2x+5y=4 \\ 3y−z=3 \\ 4x+3z=−3 \end{array} \right.\)
\((−3,2,3)\)
\(\left\{ \begin{array} {l} 2y+3z=−1 \\ 5x+3y=−6 \\ 7x+z=1 \end{array} \right.\)
\(\left\{ \begin{array} {l} 3x−z=−3 \\ 5y+2z=−6 \\ 4x+3y=−8 \end{array} \right.\)
\((−2,0,−3)\)
\(\left\{ \begin{array} {l} 2x+3y+z=1 \\ 2x+y+z=9 \\ 3x+4y+2z=20 \end{array} \right.\)
\(\left\{ \begin{array} {l} x+2y+6z=5 \\ −x+y−2z=3 \\ x−4y−2z=1 \end{array} \right.\)
no solution
\(\left\{ \begin{array} {l} x+2y−3z=−1 \\ x−3y+z=1 \\ 2x−y−2z=2 \end{array} \right.\)
\(\left\{ \begin{array} {l} 4x−3y+2z=0 \\ −2x+3y−7z=1 \\ 2x−2y+3z=6 \end{array} \right.\)
\(\left\{ \begin{array} {l} x−y+2z=−4 \\ 2x+y+3z=2 \\ −3x+3y−6z=12 \end{array} \right.\)
\(\left\{ \begin{array} {l} −x−3y+2z=14 \\ −x+2y−3z=−4 \\ 3x+y−2z=6 \end{array} \right.\)
infinitely many solutions \((x,y,z)\) where \(x=12z+4;\space y=12z−6;\space z\) is any real number
\(\left\{ \begin{array} {l} x+y−3z=−1 \\ y−z=0 \\ −x+2y=1 \end{array} \right.\)
\(\left\{ \begin{array} {l} x+2y+z=4 \\ x+y−2z=3 \\ −2x−3y+z=−7 \end{array} \right.\)
infinitely many solutions \((x,y,z)\) where \(x=5z+2;\space y=−3z+1;\space z\) is any real number
Writing Exercises
Solve the system of equations \(\left\{ \begin{array} {l} x+y=10 \\ x−y=6\end{array} \right.\) ⓐ by graphing and ⓑ by substitution. ⓒ Which method do you prefer? Why?
Solve the system of equations \(\left\{ \begin{array} {l} 3x+y=1 \\ 2x=y−8 \end{array} \right.\) by substitution and explain all your steps in words.
Answers will vary.
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
![matrices and row operations assignment quizlet This table has 4 columns 5 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don’t get it. The first column has the following statements: Write the augmented matrix for a system of equations, Use row operations on a matrix, Solve systems of equations using matrices, Write the augmented matrix for a system of equations, Use row operations on a matrix. The remaining columns are blank.](https://math.libretexts.org/@api/deki/files/20398/CNX_IntAlg_Figure_04_05_201_img_new.jpg?revision=1)
ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?
Module 14: Solve Systems With Matrices
Row operations and augmented matrices, learning outcomes.
- Write the augmented matrix for a system of equations.
- Perform row operations on an augmented matrix.
A matrix can serve as a device for representing and solving a system of equations. To express a system in matrix form, we extract the coefficients of the variables and the constants, and these become the entries of the matrix. We use a vertical line to separate the coefficient entries from the constants, essentially replacing the equal signs. When a system is written in this form, we call it an augmented matrix .
For example, consider the following [latex]2\times 2[/latex] system of equations.
We can write this system as an augmented matrix:
We can also write a matrix containing just the coefficients. This is called the coefficient matrix .
A three-by-three system of equations such as
has a coefficient matrix
and is represented by the augmented matrix
Notice that the matrix is written so that the variables line up in their own columns: x -terms go in the first column, y -terms in the second column, and z -terms in the third column. It is very important that each equation is written in standard form [latex]ax+by+cz=d[/latex] so that the variables line up. When there is a missing variable term in an equation, the coefficient is 0.
How To: Given a system of equations, write an augmented matrix
- Write the coefficients of the x -terms as the numbers down the first column.
- Write the coefficients of the y -terms as the numbers down the second column.
- If there are z -terms, write the coefficients as the numbers down the third column.
- Draw a vertical line and write the constants to the right of the line.
Example: Writing the Augmented Matrix for a System of Equations
Write the augmented matrix for the given system of equations.
The augmented matrix displays the coefficients of the variables and an additional column for the constants.
Write the augmented matrix of the given system of equations.
[latex]\left[\begin{array}{rr}\hfill 4& \hfill -3 \\ \hfill 3& \hfill 2\end{array}\text{ }|\text{ }\begin{array}{r}\hfill 11\\ \hfill 4\end{array}\right][/latex]
Writing a System of Equations from an Augmented Matrix
We can use augmented matrices to help us solve systems of equations because they simplify operations when the systems are not encumbered by the variables. However, it is important to understand how to move back and forth between formats in order to make finding solutions smoother and more intuitive. Here, we will use the information in an augmented matrix to write the system of equations in standard form.
Example: Writing a System of Equations from an Augmented Matrix Form
Find the system of equations from the augmented matrix.
Write the system of equations from the augmented matrix.
Row Operations
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.
We use row operations corresponding to equation operations to obtain a new matrix that is row-equivalent in a simpler form. Here are the guidelines to obtaining row-echelon form.
- 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 of 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.
To solve a system of equations we can perform the following row operations to convert the coefficient matrix to row-echelon form and do back-substitution to find the solution.
- 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])
Each of the row operations corresponds to the operations we have already learned to solve systems of equations in three variables. With these operations, there are some key moves that will quickly achieve the goal of writing a matrix in row-echelon form. To obtain a matrix in row-echelon form for finding solutions, we use Gaussian elimination, a method that uses row operations to obtain a 1 as the first entry so that row 1 can be used to convert the remaining rows.
A General Note: Gaussian Elimination
The Gaussian elimination method refers to a strategy used to obtain the row-echelon form of a matrix. The goal is to write matrix [latex]A[/latex] with the number 1 as the entry down the main diagonal and have all zeros below.
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: Performing Row Operations on a 3×3 Augmented Matrix to Obtain Row-Echelon Form
Perform row operations on the given matrix to obtain row-echelon form.
The first row already has a 1 in row 1, column 1. The next step is to multiply row 1 by [latex]-2[/latex] and add it to row 2. Then replace row 2 with the result.
Next, obtain a zero in row 3, column 1.
Next, obtain a zero in row 3, column 2.
The last step is to obtain a 1 in row 3, column 3.
Write the system of equations in row-echelon form.
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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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what are the three elementary row operations. 1) replacement. 2) interchange. 3) scaling. if a matrix has all zeros for the bottom row, except for the last position, what does this mean? there is no solution. coefficient matrix. the matrix containing only the coefficients of the system. augmented matrix.
1. If a row does not consist entirely of zeros, then the first nonzero number in the row is a leading 1 2. If there are any rows that consist entirely of zeros, then they are grouped together at the bottom of the matrix 3. In any two successive rows that do not consist entirely of zeros, the leading 1 in the lower row occurs farther on the right than the leading 1 in the higher row 4.
A matrix with the same number of columns and rows. Determinant of a 2 X 2 matrix. The determinant of a 2 X 2 matrix is ad - bc. Inverse of a 2 X 2 matrix. The inverse matrix equals 1/detA times {d, -b, -c, a}. Study with Quizlet and memorize flashcards containing terms like Matrix, Matrix element, Matrix addition and more.
Definition: MATRICES. A matrix is a rectangular array of numbers that is usually named by a capital letter: , , ,and so on. Each entry in a matrix is referred to as ,such that represents the row and represents the column. Matrices are often referred to by their dimensions: indicating rows and columns.
To indicate this matrix row operation, we can write the following: [ 2 3 4 0 8 1 ] → R 1 + R 2 → R 2 [ 2 3 4 2 11 5 ] Notice how the sum of row 1 and 2 replaces row 2 .
Matrix row operations. Perform the row operation, R 1 ↔ R 2 , on the following matrix. Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.
Key Idea \ (\PageIndex {1}\): Elementary Row Operations. Add a scalar multiple of one row to another row, and replace the latter row with that sum. Multiply one row by a nonzero scalar. Swap the position of two rows. Given any system of linear equations, we can find a solution (if one exists) by using these three row operations.
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.
Here are the guidelines to obtaining row-echelon form. 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.
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.
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.
A square matrix is a matrix with dimensions n×n,n×n, meaning that it has the same number of rows as columns. The 3×33×3 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. [a11a12a13][ a11a12a13 ] A column matrix is a matrix consisting of one column with dimensions m×1.m×1.
Use Row Operations on a Matrix. In the following exercises, perform the indicated operations on the augmented matrices. [ 6 3 −4 −2 3 1] [ 6 − 4 3 3 − 2 1] ⓐ Interchange rows 1 and 2. ⓑ Multiply row 2 by 3. ⓒ Multiply row 2 by −2 − 2 and add row 1 to it. [ 4 3 −6 2 −3 1] [ 4 − 6 − 3 3 2 1] ⓐ Interchange rows 1 and 2.
The only m by m echelon matrix is the identity matrix. Therefore every invertible matrix is row-equivalent to the identity. One consequence of this is that the inverse of A can be found by row operations, as follows: 4.1 Inverting a matrix by row operations Write down A next to I as an m by 2m matrix (A;I). Perform row operations on this in such
Multiply row two by the scalar −7. Add that product to row three. Then, place the result in the new position of row three. • Row echelon form is when all elements of a matrix below the main diagonal are 0. • Notice, in the linear system, the variable z has already been isolated.
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.
Matrices and Row Operations. magine earning . 10. Summary. LESSON. Question. How can you manipulate rows in matrices to solve problems? Answer (Sample answer) Rows can be switched, multiplied by a scalar, or added to one another to get the matrix in row echelon form. To solve for the variables, use the reduced row echelon form of the matrix ...
Combining Multiplication and Addition. Combining multiplication and addition is also called a linear combination. Multiply. row two by the scalar - 7. Add that product to row three. Then, place the result in. the new position of row three. 4 −4 12152 0 1 −1ቮ−12 0 7 −6−73. - 7 2 + 3 → 3.
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.