Write each matrix equation as a system of linear equations without matrices.

$\begin{bmatrix} 4 & -7 \\ 2 & -3 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} -3 \\ 1 \end{bmatrix}$

Write each matrix equation as a system of linear equations without matrices.

$\begin{bmatrix} 2 & 0 & -1 \\ 0 & 3 & 0 \\ 1 & 1 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 6 \\ 9 \\ 5 \end{bmatrix}$

a. Write each linear system as a matrix equation in the form AX = B. b. Solve the system using the inverse that is given for the coefficient matrix.

$\begin{cases} 2x + 6y + 6z = 8 \\ 2x + 7y + 6z = 10 \\ 2x + 7y + 7z = 9 \end{cases} \\ \text{The inverse of } \begin{bmatrix} 2 & 6 & 6 \\ 2 & 7 & 6 \\ 2 & 7 & 7 \end{bmatrix} \text{ is } \begin{bmatrix} \frac{7}{2} & 0 & -3 \\ -1 & 0 & 0 \\ 0 & -1 & 1 \end{bmatrix}\text{.}$

a. Write each linear system as a matrix equation in the form AX = B. b. Solve the system using the inverse that is given for the coefficient matrix.

$\begin{cases} x - y + z = 8 \\ 2y - z = -7 \\ 2x + 3y = 1 \end{cases} \\ \text{The inverse of } \begin{bmatrix} 1 & -1 & 1 \\ 0 & 2 & -1 \\ 2 & 3 & 0 \end{bmatrix} \text{ is } \begin{bmatrix} 3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2 \end{bmatrix}\text{.}$

a. Write each linear system as a matrix equation in the form AX = B. b. Solve the system using the inverse that is given for the coefficient matrix.

$\begin{cases} w - x + 2y \quad\quad = -3 \\ \quad\quad x - y + z = 4 \\ -w + x - y + 2z = 2 \\ \quad\quad -x + y - 2z = -4 \end{cases} \\ \text{The inverse of } \begin{bmatrix} 1 & -1 & 2 & 0 \\ 0 & 1 & -1 & 1 \\ -1 & 1 & -1 & 2 \\ 0 & -1 & 1 & -2 \end{bmatrix} \text{ is } \begin{bmatrix} 0 & 0 & -1 & -1 \\ 1 & 4 & 1 & 3 \\ 1 & 2 & 1 & 2 \\ 0 & -1 & 0 & -1 \end{bmatrix}$

Evaluate each determinant in Exercises 1–10.

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

Evaluate each determinant in Exercises 1–10.

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

Evaluate each determinant in Exercises 1–10.

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

Evaluate each determinant in Exercises 1–10.

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

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

For Exercises 11–22, use Cramer's Rule to solve each system. $\begin{cases} x + y = 7 \\ x - y = 3 \end{cases}$

For Exercises 11–22, use Cramer's Rule to solve each system. $\begin{cases} 12x + 3y = 15 \\ 2x - 3y = 13 \end{cases}$

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

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

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

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

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

In Exercises 23–30, use expansion by minors to evaluate each determinant. $\begin{vmatrix} 3 & 1 & 0 \\ -3 & 4 & 0 \\ -1 & 3 & -5 \end{vmatrix}$

In Exercises 23–30, use expansion by minors to evaluate each determinant. $\begin{vmatrix} 1 & 1 & 1 \\ 2 & 2 & 2 \\ -3 & 4 & -5 \end{vmatrix}$

In Exercises 23–30, use expansion by minors to evaluate each determinant. $\begin{vmatrix} 0.5 & 7 & 5 \\ 0.5 & 3 & 9 \\ 0.5 & 1 & 3 \end{vmatrix}$

In Exercises 31–36, use the alternative method for evaluating third-order determinants on here to evaluate each determinant. $\begin{vmatrix} -3 & 4 & -5 \\ 5 & -2 & 0 \\ 8 & -1 & 3 \end{vmatrix}$

In Exercises 31–36, use the alternative method for evaluating third-order determinants on here to evaluate each determinant. $\begin{vmatrix} 1 & 5 & 6 \\ 1 & 4 & 5 \\ 1 & 9 & 10 \end{vmatrix}$

In Exercises 31–36, use the alternative method for evaluating third-order determinants on here to evaluate each determinant. $\begin{vmatrix} 0.5 & 7 & 5 \\ 0.5 & 3 & 9 \\ 0.5 & 1 & 3 \end{vmatrix}$

In Exercises 37–44, use Cramer's Rule to solve each system. $\begin{cases} x + y + z = 0 \\ 2x - y + z = -1 \\ -x + 3y - z = -8 \end{cases}$

In Exercises 37–44, use Cramer's Rule to solve each system. $\begin{cases} 4x - 5y - 6z = -1 \\ x - 2y - 5z = -12 \\ 2x - y = 7 \end{cases}$

In Exercises 37–44, use Cramer's Rule to solve each system. $\begin{cases} x + y + z = 4 \\ x - 2y + z = 7 \\ x + 3y + 2z = 4 \end{cases}$

In Exercises 37–44, use Cramer's Rule to solve each system. $\begin{cases} x + 2z = 4 \\ 2y - z = 5 \\ 2x + 3y = 13 \end{cases}$

Evaluate each determinant in Exercises 49–52. $\begin{vmatrix} 4 & 2 & 8 & -7 \\ -2 & 0 & 4 & 1 \\ 5 & 0 & 0 & 5 \\ 4 & 0 & 0 & -1 \end{vmatrix}$

Evaluate each determinant in Exercises 49–52. $\begin{vmatrix} -2 & -3 & 3 & 5 \\ 1 & -4 & 0 & 0 \\ 1 & 2 & 2 & -3 \\ 2 & 0 & 1 & 1 \end{vmatrix}$

In Exercises 53–54, evaluate each determinant. $\begin{vmatrix} \begin{vmatrix} 3 & 1 \\ -2 & 3 \end{vmatrix} & \begin{vmatrix} 7 & 0 \\ 1 & 5 \end{vmatrix} \\ \begin{vmatrix} 3 & 0 \\ 0 & 7 \end{vmatrix} & \begin{vmatrix} 9 & -6 \\ 3 & 5 \end{vmatrix} \end{vmatrix}$