7. Systems of Equations & Matrices

Use the determinant theorems to evaluate each determinant.

Use the determinant theorems to evaluate each determinant. See Example 4.

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7.

x + y = 4

2x - y = 2

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7.

4x + 3y = -7

2x + 3y = -11

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7.

$5x+4y=10$

$5x+4y=10$

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7. 1.5

x + 3y = 5

2x + 4y = 3

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7.

(1/2)x + (1/3)y = 2

(3/2)x - (1/2)y = -12

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7.

2x - y + 4z = -2

3x + 2y - z = -3

x + 4y - 2z = 17

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7.

x + 2y + 3z = 4

4x + 3y + 2z = 1

-x - 2y - 3z = 0

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7.

-2x - 2y + 3z = 4

5x + 7y - z = 2

2x + 2y - 3z = -4

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

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

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

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.

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