7. Systems of Equations & Matrices

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.

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

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.

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

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.

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

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

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

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

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

Use the fact that if , then to find the inverse of each matrix, if possible. Check that $A{A}^{-1}=I_2$ and $A^{-1}A=I_2$.

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