Blitzer - College Algebra 8th Edition

Ch. 6 - Matrices and Determinants

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Blitzer 8th Edition

Ch. 6 - Matrices and Determinants

Problem 15

Use the fact that if $A = [\begin{matrix} a & b \\ c & d \end{matrix}]$ , then $A^{- 1} = \frac{1}{a d - b c} [\begin{matrix} d & - b \\ - c & a \end{matrix}]$ to find the inverse of each matrix, if possible. Check that $A A^{- 1} = I_{2}$ and $A^{- 1} A = I_{2}$ .

$A = \begin{bmatrix} 3 & -1 \\ -4 & 2 \end{bmatrix}$

Problem 15

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. $\begin{cases} 2x + y - z = 2 \\ 3x + 3y - 2z = 3 \end{cases}$

Problem 15

Perform each matrix row operation and write the new matrix.

$\begin{bmatrix} 1 & -3 & 2 & \vert & 0 \\ 3 & 1 & -1 & \vert & 7 \\ 2 & -2 & 1 & \vert & 3 \end{bmatrix} \quad -3R_1 + R_2$

Problem 15

For Exercises 11–22, use Cramer's Rule to solve each system. $\begin{cases} 4x - 5y = 17 \\ 2x + 3y = 3 \end{cases}$

Problem 15a

In Exercises 9 - 16, find the following matrices: a. A + B

Problem 15b

In Exercises 9 - 16, find the following matrices: b. A - B

Problem 15c

In Exercises 9 - 16, find the following matrices: c. - 4A

Problem 15d

In Exercises 9 - 16, find the following matrices: d. - 3A + 2B

Problem 16

In Exercises 14–27, perform the indicated matrix operations given that and D are defined as follows. If an operation is not defined, state the reason. D-A

Problem 17

$A = \begin{bmatrix} 10 & -2 \\ -5 & 1 \end{bmatrix}$

Problem 17

Let $A = [\begin{matrix} - 3 & - 7 \\ 2 & - 9 \\ 5 & 0 \end{matrix}]$ and $B = [\begin{matrix} - 5 & - 1 \\ 0 & 0 \\ 3 & - 4 \end{matrix}]$ Solve each matrix equation for X. X - A = B

Problem 17

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. $\begin{cases} x + 2y + 3z = 5 \\ y - 5z = 0 \end{cases}$

Problem 17

For Exercises 11–22, use Cramer's Rule to solve each system. $\begin{cases} x + 2y = 3 \\ 3x - 4y = 4 \end{cases}$

Problem 17

Perform each matrix row operation and write the new matrix.

$\begin{bmatrix} 1 & -1 & 1 & 1 & \vert & 3 \\ 0 & 1 & -2 & -1 & \vert & 0 \\ 2 & 0 & 3 & 4 & \vert & 11 \\ 5 & 1 & 2 & 4 & \vert & 6 \end{bmatrix} \quad \begin{array}{l} -2R_1 + R_3 \\ -5R_1 + R_4 \end{array}$

Problem 18

In Exercises 14–27, perform the indicated matrix operations given that and D are defined as follows. If an operation is not defined, state the reason. 3A+2D

Problem 19

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. $\begin{cases} x + y - 2z = 2 \\ 3x - y - 6z = -7 \end{cases}$

Problem 19

Let $A = [\begin{matrix} - 3 & - 7 \\ 2 & - 9 \\ 5 & 0 \end{matrix}]$ and $B = [\begin{matrix} - 5 & - 1 \\ 0 & 0 \\ 3 & - 4 \end{matrix}]$ . Solve each matrix equation for X. 2X + A = B

Problem 19

For Exercises 11–22, use Cramer's Rule to solve each system. $\begin{cases} 3x - 4y = 4 \\ 2x + 2y = 12 \end{cases}$

Problem 19

In Exercises 19–20, a few steps in the process of simplifying the given matrix to row-echelon form, with 1s down the diagonal from upper left to lower right, and 0s below the 1s, are shown. Fill in the missing numbers in the steps that are shown.

Problem 20

In Exercises 14–27, perform the indicated matrix operations given that and D are defined as follows. If an operation is not defined, state the reason. -5(A+D)

Problem 21

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

$\begin{cases} x + y - z = -2 \\ 2x - y + z = 5 \\ -x + 2y + 2z = 1 \end{cases}$

Problem 21

Let $A = [\begin{matrix} - 3 & - 7 \\ 2 & - 9 \\ 5 & 0 \end{matrix}]$ and $B = [\begin{matrix} - 5 & - 1 \\ 0 & 0 \\ 3 & - 4 \end{matrix}]$ . Solve each matrix equation for X. 3X + 2A = B

Problem 21

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. $\begin{cases} w + x - y + z = -2 \\ 2w - x + 2y - z = 7 \\ -w + 2x + y + 2z = -1 \end{cases}$

Problem 21

For Exercises 11–22, use Cramer's Rule to solve each system. $\begin{cases} 2x = 3y + 2 \\ 5x = 51 - 4y \end{cases}$

Problem 23

Perform the indicated matrix operations given that and D are defined as follows. If an operation is not defined, state the reason. BD

$A = [\begin{matrix} 2 & - 1 & 2 \\ 5 & 3 & - 1 \end{matrix}]$

Problem 23

In Exercises 23–30, use expansion by minors to evaluate each determinant. $\begin{vmatrix} 3 & 0 & 0 \\ 2 & 1 & -5 \\ 2 & 5 & -1 \end{vmatrix}$

Problem 23

Let $A = [\begin{matrix} - 3 & - 7 \\ 2 & - 9 \\ 5 & 0 \end{matrix}]$ and $B = [\begin{matrix} - 5 & - 1 \\ 0 & 0 \\ 3 & - 4 \end{matrix}]$ . Solve each matrix equation for X. B - X = 4A

Problem 23

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

$\begin{cases} x + 3y = 0 \\ x + y + z = 1 \\ 3x - y - z = 11 \end{cases}$

Problem 23

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. $\begin{cases} w + 2x + 3y - z = 7 \\ 2x - 3y + z = 4 \\ w - 4x + y = 3 \end{cases}$

Problem 25

Let $A = [\begin{matrix} - 3 & - 7 \\ 2 & - 9 \\ 5 & 0 \end{matrix}]$ and $B = [\begin{matrix} - 5 & - 1 \\ 0 & 0 \\ 3 & - 4 \end{matrix}]$ . Solve each matrix equation for X. 4A + 3B = - 2X