Blitzer - College Algebra 8th Edition

Ch. 6 - Matrices and Determinants

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Blitzer 8th Edition

Ch. 6 - Matrices and Determinants

Problem 1

a. Give the order of each matrix.

b. If $A = [a_{i j}]$ , identify $a sub 32$ and $a sub 23$ , or explain why identification is not possible.

$\begin{bmatrix} 4 & -7 & 5 \\ -6 & 8 & -1 \end{bmatrix}$

Problem 1

Find the products AB and BA to determine whether B is the multiplicative inverse of A.

$A = \begin{bmatrix} 4 & -3 \\ -5 & 4 \end{bmatrix} , B = \begin{bmatrix} 4 & 3 \\ 5 & 4 \end{bmatrix}$

Problem 1

Write the augmented matrix for each system of linear equations.

$\begin{cases} 2x + y + 2z = 2 \\ 3x - 5y - z = 4 \\ x - 2y - 3z = -6 \end{cases}$

Problem 1

Perform each matrix row operation and write the new matrix.

$\begin{bmatrix} 1 & 2 & 2 & \vert & 2 \\ 0 & 1 & -1 & \vert & 2 \\ 0 & 5 & 4 & \vert & 1 \end{bmatrix} -5R_2 + R_3$

Problem 1

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. $\begin{cases} 5x + 12y + z = 10 \\ 2x + 5y + 2z = -1 \\ x + 2y - 3z = 5 \end{cases}$

Problem 1

Evaluate each determinant in Exercises 1–10.

$\begin{vmatrix} 5 & 7 \\ 2 & 3 \end{vmatrix}$

Problem 3

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

${\begin{matrix} x + 2 y + 3 z = - 5 \\ 2 x + y + z = 1 \\ x + y - z = 8 \end{matrix}$

Problem 3

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. $\begin{cases} 5x + 8y - 6z = 14 \\ 3x + 4y - 2z = 8 \\ x + 2y - 2z = 3 \end{cases}$

Problem 3

Evaluate each determinant in Exercises 1–10.

$\begin{vmatrix} -4 & 1 \\ 5 & 6 \end{vmatrix}$

Problem 3

Find the products AB and BA to determine whether B is the multiplicative inverse of A.

$A = [\begin{matrix} - 4 & 0 \\ 1 & 3 \end{matrix}],$

Problem 3

a. Give the order of each matrix.

b. If A = [aᵢⱼ] , identify a₃₂ and a₂₃, or explain why identification is not possible.

$\begin{bmatrix} 1 & -5 & \pi & e \\ 0 & 7 & -6 & -\pi \\ -2 & \frac{1}{2} & 11 & -\frac{1}{5} \end{bmatrix}$

Problem 3

Write the augmented matrix for each system of linear equations.

$\begin{cases} x - y + z = 8 \\ y - 12z = -15 \\ z = 1 \end{cases}$

Problem 4

In Exercises 3–5, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

Problem 5

In Exercises 5 - 8, find values for the variables so that the matrices in each exercise are equal. $\begin{bmatrix} x \\ 4 \end{bmatrix} = \begin{bmatrix} 6 \\ y \end{bmatrix}$

Problem 5

Evaluate each determinant in Exercises 1–10.

$\begin{vmatrix} -7 & 14 \\ 2 & -4 \end{vmatrix}$

Problem 5

Find the products AB and BA to determine whether B is the multiplicative inverse of A.

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

Problem 5

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. $\begin{cases} 3x + 4y + 2z = 3 \\ 4x - 2y - 8z = -4 \\ x + y - z = 3 \end{cases}$

Problem 5

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

${\begin{matrix} 3 x_{1} + 5 x_{2} - 8 x_{3} + 5 x_{4} = - 8 \\ x_{1} + 2 x_{2} - 3 x_{3} + x_{4} = - 7 \\ 2 x_{1} + 3 x_{2} - 7 x_{3} + 3 x_{4} = - 11 \\ 4 x_{1} + 8 x_{2} - 10 x_{3} + 7 x_{4} = - 10 \end{matrix}$

Problem 5

Write the augmented matrix for each system of linear equations.

$\begin{cases} 5x - 2y - 3z = 0 \\ x + y = 5 \\ 2x - 3z = 4 \end{cases}$

Problem 7

In Exercises 5 - 8, find values for the variables so that the matrices in each exercise are equal. $\begin{bmatrix} x & 2y \\ z & 9 \end{bmatrix} = \begin{bmatrix} 4 & 12 \\ 3 & 9 \end{bmatrix}$

Problem 7

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. $\begin{cases} 8x + 5y + 11z = 30 \\ -x - 4y + 2z = 3 \\ 2x - y + 5z = 12 \end{cases}$

Problem 7

Write the augmented matrix for each system of linear equations.

$\begin{cases} 2w + 5x - 3y + z = 2 \\ 3x + y = 4 \\ w - x + 5y = 9 \\ 5w - 5x - 2y = 1 \end{cases}$

Problem 7

Evaluate each determinant in Exercises 1–10.

$\begin{vmatrix} -5 & -1 \\ -2 & -7 \end{vmatrix}$

Problem 7

Find the products AB and BA to determine whether B is the multiplicative inverse of A.

$A = [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{matrix}],$

Problem 8

In Exercises 8–11, use Gaussian elimination to find the complete solution to each system, or show that none exists.

Problem 9

In Exercises 8–11, use Gaussian elimination to find the complete solution to each system, or show that none exists.

Problem 9

Find the products AB and BA to determine whether B is the multiplicative inverse of A.

$A = [\begin{matrix} 1 & 2 & 3 \\ 1 & 3 & 4 \\ 1 & 4 & 3 \end{matrix}],$

Problem 9

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. $\begin{cases} w - 2x - y - 3z = -9 \\ w + x - y = 0 \\ 3w + 4x + z = 6 \\ 2x - 2y + z = 3 \end{cases}$

Problem 9

Evaluate each determinant in Exercises 1–10. $\begin{vmatrix} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{8} & - \frac{3}{4} \end{vmatrix}$

Problem 9

Write the system of linear equations represented by the augmented matrix. Use x, y, and z, or, if necessary, w, x, y, and z, for the variables.

$\begin{bmatrix} 5 & 0 & 3 & \vert & -11 \\ 0 & 1 & -4 & \vert & 12 \\ 7 & 2 & 0 & \vert & 3 \end{bmatrix}$