7. Systems of Equations & Matrices

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

Evaluate each determinant.

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

$\(\begin{vmatrix}\)-4 & 1 & 4\\2& 0 & 1\\ 0&2&4\(\end{vmatrix}\)$

Evaluate each determinant.

For each pair of matrices A and B, find (a) AB and (b) BA. See Example 7.

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\(\end{vmatrix}\)$

Find AB and BA for the following matrices.

Matrix B acts as the multiplicative element for 2 $\times$ 2 square matrices.

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

Evaluate the determinant of the matrix.

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

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

Find each sum or difference, if possible. See Examples 2 and 3.

Evaluate each determinant.

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

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