7. Systems of Equations & Matrices

In Exercises 39–42, find A^(-1) Check that AA^-1 = I and A^(-1)A = I

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\(\end{vmatrix}\)$

Evaluate each determinant.

Answer each question. What is the product ?

Solve each equation.

Find each product, if possible.

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\(\end{bmatrix}\)$

Find the inverse, if it exists, for each matrix.

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

Find each product, if possible.

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\(\end{cases}\)$

If the dimension of matrix A is 3 $\times$ 2 and the dimension of matrix B is 2 $\times$ 6, then the dimension of AB is ____.

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\(\end{vmatrix}\)$

In Exercises 37–38, find the products and to determine whether B is the multiplicative inverse of A.