7. Systems of Equations & Matrices

Evaluate the determinant of the matrix.

Find the cofactor of each element in the second row of each matrix.

For each pair of matrices A and B, find (a) AB and (b) BA.

Let and . Find each of the following. See Examples 2 –4.

(3/2)B

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

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

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{.}\)$

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

Evaluate each determinant.

Let and . Find each of the following. See Examples 2 –4.

2A + 4B

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

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

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\(\end{vmatrix}\)$

Find each sum or difference, if possible.