7. Systems of Equations & Matrices

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. See Example 4.

Find the dimension of each matrix. Identify any square, column, or row matrices. See the discussion preceding Example 1.

$\mid \begin{array}{c}-9\end{array}\mid$

Perform each operation, if possible.

Find the values of the variables for which each statement is true, if possible.

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 + y = 4

2x - y = 2

In Exercises 37–44, use Cramer's Rule to solve each system. $\(\begin{cases}\)4x - 5y - 6z = -1 \\(\x\) - 2y - 5z = -12 \\2x - y = 7\(\end{cases}\)$

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.

$5x+4y=10$

$5x+4y=10$

In Exercises 57–60, solve each equation for x.

In Exercises 43–44, (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.

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

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

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

Evaluate each determinant.