7. Systems of Equations & Matrices

Evaluate each determinant.

Evaluate each determinant.

For Exercises 11–22, use Cramer's Rule to solve each system. $\(\begin{cases}\)x + y = 7 \\(\x\) - y = 3\(\end{cases}\)$

Write each linear system as a matrix equation in the form AX = B, where A is the coefficient matrix and B is the constant matrix.

$\(\begin{cases}\)x + 3y + 4z = -3 \\(\x\) + 2y + 3z = -2 \\(\x\) + 4y + 3z = -6\(\end{cases}\)$

In Exercises 53–54, evaluate each determinant. $\(\begin{vmatrix}\[\begin{vmatrix}\) 3 & 1 \\ -2 & 3 \(\end{vmatrix}\) & \(\begin{vmatrix}\) 7 & 0 \\ 1 & 5 \(\end{vmatrix}\) \\\(\begin{vmatrix}\) 3 & 0 \\ 0 & 7 \(\end{vmatrix}\) & \(\begin{vmatrix}\) 9 & -6 \\ 3 & 5 \(\end{vmatrix}\]\end{vmatrix}\)$

Use the determinant theorems to evaluate each determinant.

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

Evaluate each determinant.

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

In Exercises 37–44, use Cramer's Rule to solve each system. $\(\begin{cases}\)x + y + z = 0 \\2x - y + z = -1 \\-x + 3y - z = -8\(\end{cases}\)$

Perform each operation, if possible.

$\left[\begin{array}{c}3\\ 2\\ 5\end{array}\right]-\left[\begin{array}{c}8\\ -4\\ 6\end{array}\right]+\left[\begin{array}{c}1\\ 0\\ 2\end{array}\right]$

Perform each operation, if possible.

Use the determinant theorems to evaluate each determinant.

Evaluate each determinant.

In Exercises 23–30, use expansion by minors to evaluate each determinant. $\(\begin{vmatrix}\)3 & 1 & 0 \\-3 & 4 & 0 \\-1 & 3 & -5\(\end{vmatrix}\)$