7. Systems of Equations & Matrices

In Exercises 37–44, use Cramer's Rule to solve each system. $\begin{cases}x + y + z = 4 \\x - 2y + z = 7 \\x + 3y + 2z = 4\end{cases}$

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

In Exercises 45–48, explain why the system of equations cannot be solved using Cramer's Rule. Then use Gaussian elimination to solve the system.

In Exercises 46–51, evaluate each determinant.

In Exercises 45–48, explain why the system of equations cannot be solved using Cramer's Rule. Then use Gaussian elimination to solve the system.

Use the determinant theorems to evaluate each determinant.

Solve each equation.

Solve each equation.

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

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 in Exercises 49–52. $\begin{vmatrix}-2 & -3 & 3 & 5 \\1 & -4 & 0 & 0 \\1 & 2 & 2 & -3 \\2 & 0 & 1 & 1\end{vmatrix}$

Evaluate each determinant.

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

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