7. Systems of Equations & Matrices

Introduction to Matrices

7. Systems of Equations & Matrices

# Introduction to Matrices - Video Tutorials & Practice Problems

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concept

## Introduction to Matrices

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Problem

ProblemWrite the equations in standard form, then represent the system using an augmented matrix**.**

$3x+5y-9=0$

$8x=-4y+3$

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B

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D

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Problem

ProblemWrite the system of equations represented by the augmented matrix shown**.**

****

A

$x+2y+3z=5;5y+4z=1;4x+7y=12$

B

$x+2y+3x=5;5x+4y=1;4x+7y=12$

C

$x+2y+3x=5;5y+4z=1;4y+7z=12$

D

$x+2y+3x=-5;5y+4z=-1;4x+7y=-12$

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concept

## Performing Row Operations on Matrices

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Problem

ProblemPerform the indicated **Row Operation.**

**SWAP **$R_1\leftrightarrow R_2$

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B

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D

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Problem

ProblemPerform the indicated **Row Operation.**

**ADD** $R_1+2\cdot R_3\rightarrow R_1$

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B

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concept

## Solving Systems of Equations - Matrices (Row-Echelon Form)

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example

## Example 1

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example

## Example 2

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concept

## Solving Systems of Equations - Matrices (Reduced Row-Echelon Form)

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11

Problem

ProblemSolve the system of equations by using row operations to write a matrix in ** REDUCED **row-echelon form

**.**

$4x+2y+3z=6$

$x+y+z=3$

$5x+y+2z=5$

A

$x=0,y=-3,z=4$

B

$x=-\frac14,y=-\frac{19}{4},z=\frac{11}{2}$

C

$x=1,y=4,z=-2$

D

$x=\frac12,y=\frac12,z=1$

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PRACTICE PROBLEMS AND ACTIVITIES (71)

- In Exercises 1–8, write the augmented matrix for each system of linear equations.
- In Exercises 1–2, perform each matrix row operation and write the new matrix.
- How many rows and how many columns does this matrix have? What is its dimension? <4x2 Matrix>
- In Exercises 1 - 4, a. Give the order of each matrix, b. If A = [a_ij], identify a_32 and a_23, or explain why...
- In Exercises 1–8, write the augmented matrix for each system of linear equations.
- In Exercises 3–5, solve each system of equations using matrices. Use Gaussian elimination with back-substituti...
- What is the augmented matrix of the following system? -3x + 5y = 2 6x + 2y = 7
- In Exercises 3–5, solve each system of equations using matrices. Use Gaussian elimination with back-substituti...
- In Exercises 3–5, solve each system of equations using matrices. Use Gaussian elimination with back-substituti...
- In Exercises 1–8, write the augmented matrix for each system of linear equations.
- Use the given row transformation to change each matrix as indicated. See Sample 1. < 2x2 Matrix > ; -4 ...
- In Exercises 1–8, write the augmented matrix for each system of linear equations.
- Use the given row transformation to change each matrix as indicated. See Sample 1. < 2x2 Matrix > ; -7 ...
- In Exercises 9-12, write the system of linear equations represented by the augmented matrix. Use x, y, and z, ...
- Use the given row transformation to change each matrix as indicated. See Sample 1. < 3x3 Matrix > ; 2 t...
- In Exercises 9 - 16, find the following matrices: b. A - B 4 1 5 9 A = B = 3 2 0 7
- Find the dimension of each matrix. Identify any square, column, or row matrices. See the discussion preceding ...
- In Exercises 9 - 16, find the following matrices: d. - 3A + 2B 4 1 5 9 A = B = 3 2 0 7
- Use the given row transformation to change each matrix as indicated. See Sample 1. < 3x3 Matrix > ; 4 t...
- In Exercises 9-12, write the system of linear equations represented by the augmented matrix. Use x, y, and z, ...
- In Exercises 9 - 16, find the following matrices: a. A + B 3 1 1 2 - 3 6 A = B = - 1 2 5 - 3 1 - 4
- In Exercises 9 - 16, find the following matrices: d. - 3A + 2B 3 1 1 2 - 3 6 A = B = - 1 2 5 - 3 1 - 4
- In Exercises 9 - 16, find the following matrices: c. - 4A 3 1 1 2 - 3 6 A = B = - 1 2 5 - 3 1 - 4
- Write the augmented matrix for each system and give its dimension. Do not solve. 2x + y + z - 3 = 0 3x - 4y + ...
- In Exercises 13–18, perform each matrix row operation and write the new matrix.
- Write the augmented matrix for each system and give its dimension. Do not solve. 4x - 2y + 3z - 4 = 0 3x + 5y ...
- In Exercises 14–27, perform the indicated matrix operations given that and D are defined as follows. If an ope...
- In Exercises 9 - 16, find the following matrices: a. A + B A = [6 2 - 3], B = [4 - 2 3]
- In Exercises 9 - 16, find the following matrices: c. - 4A A = [6 2 - 3], B = [4 - 2 3]
- In Exercises 13–18, perform each matrix row operation and write the new matrix.
- Write the system of equations associated with each augmented matrix . Do not solve. <4x3 Matrix>
- In Exercises 9 - 16, find the following matrices: d. - 3A + 2B 2 - 10 - 2 6 10 - 2 A = 14 12 10 B = 0 - 12 ...
- In Exercises 14–27, perform the indicated matrix operations given that and D are defined as follows. If an ope...
- In Exercises 13–18, perform each matrix row operation and write the new matrix.
- Write the system of equations associated with each augmented matrix . Do not solve. <4x3 Matrix>
- In Exercises 14–27, perform the indicated matrix operations given that and D are defined as follows. If an ope...
- In Exercises 19–20, a few steps in the process of simplifying the given matrix to row-echelon form, with 1s do...
- Write the system of equations associated with each augmented matrix . Do not solve.
- Find the values of the variables for which each statement is true, if possible. See Examples 1 and 2. =
- In Exercises 17 - 26, let - 3 - 7 - 5 - 1 A = 2 - 9 and B = 0 0 5 0 3 - 4 Solve each matrix equation for X....
- Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely ma...
- In Exercises 21–38, solve each system of equations using matrices. Use Gaussian elimination with back-substitu...
- Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely ma...
- In Exercises 21–38, solve each system of equations using matrices. Use Gaussian elimination with back-substitu...
- Solve each system, using the method indicated. 5x + 2y = -10 3x - 5y = -6 (Gauss-Jordan)
- In Exercises 21–38, solve each system of equations using matrices. Use Gaussian elimination with back-substitu...
- Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely ma...
- Solve each system, using the method indicated. 3x + y = -7 x - y = -5 (Gaussian elimination)
- Solve for X in the matrix equation 3X+A = B where
- Solve each system, using the method indicated. x - z = -3 y + z = 6 2x - 3z = -9 (Gauss-Jordan)
- In Exercises 27 - 36, find (if possible) the following matrices: a. AB b. BA 1 2 A = [1 2 3 4], B = 3 4
- In Exercises 27 - 36, find (if possible) the following matrices: a. AB b. BA 4 2 2 3 4 A = 6 1 B = 3 5 - 1 ...
- Find the quadratic function f(x) = ax² + bx + c for which ƒ( − 2) = −4, ƒ(1) = 2, and f(2) = 0.
- Find each sum or difference, if possible. See Examples 2 and 3. <1x4 Matrix> - <1x4 Matrix>
- Find the cubic function f(x) = ax³ + bx² + cx + d for which ƒ( − 1) = 0, ƒ(1) = 2, ƒ(2) = 3, and ƒ(3) = 12.
- Solve the system: (Hint: Let A = ln w, B = ln x, C = ln y, and D = ln z. Solve the system for A, B, C, and D....
- In Exercises 37 - 44, perform the indicated matrix operations given that A, B and C are defined as follows. If...
- Let A = and B = . Find each of the following. See Examples 2 –4. (3/2)B
- Find each product, if possible. See Examples 5–7. <4x2 Matrix>
- Find the values of the variables for which each statement is true, if possible. [2x2 matrix] = [2x2 matrix]