7. Systems of Equations & Matrices

Evaluate each determinant in Exercises 49–52. $\(\begin{vmatrix}\)4 & 2 & 8 & -7 \\-2 & 0 & 4 & 1 \\5 & 0 & 0 & 5 \\4 & 0 & 0 & -1\(\end{vmatrix}\)$

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

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

Find each product, if possible.

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 determinant theorems to evaluate each determinant.

Find each product, if possible.

Evaluate each determinant.

Solve each equation.

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.

2x - y + 4z = -2

3x + 2y - z = -3

x + 4y - 2z = 17

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

In Exercises 37–44, use Cramer's Rule to solve each system. $\(\begin{cases}\)x + 2z = 4 \\2y - z = 5 \\2x + 3y = 13\(\end{cases}\)$

Write each equation in standard form and use Cramer's Rule to solve the system.

$y-9x=-3$

$-3x=4y-1$

Answer each question. What is the product of and I₂ (in either order)?