7. Systems of Equations & Matrices

Use the determinant theorems to evaluate each determinant.

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

$A = \(\begin{bmatrix}\)3 & -1 \\-4 & 2\(\end{bmatrix}\)$

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

$y=-3x+4$

$-2x=7y-9$

Evaluate each determinant in Exercises 1–10.

$\(\begin{vmatrix}\)-4 & 1 \\5 & 6\(\end{vmatrix}\)$

What is the value of ?

In Exercises 46–51, evaluate each determinant.

Find the cofactor of each element in the second row of each matrix.

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.

-2x - 2y + 3z = 4

5x + 7y - z = 2

2x + 2y - 3z = -4

Find each sum or difference, if possible.

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

Are the given matrices inverses of each other? (Hint: Check to see whether their products are the identity matrix I_n.)

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.

Solve the system of equations using Cramer's Rule.

$4x+2y+3z=6$

$x+y+z=3$

$5x+y+2z=5$