7. Systems of Equations & Matrices

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 = \(\begin{bmatrix}\)10 & -2 \\-5 & 1\(\end{bmatrix}\)$

Find each product, if possible.

In Exercises 55–56, write the system of linear equations for which Cramer's Rule yields the given determinants.

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.

$\(\begin{cases}\)6x + 5y = 13 \\5x + 4y = 10\(\end{cases}\)$

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

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.

(1/2)x + (1/3)y = 2

(3/2)x - (1/2)y = -12

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

Let and $B = \(\left\)[ \(\begin{matrix}\) -6 & 2 \\ 4 & 0 \(\end{matrix}\) \(\right\)]$. Find each of the following.

-A + (1/2)B

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

Given and $C = \(\left\)[ \(\begin{matrix}\) -5 & 4 & 1 \\ 0 & 3 & 6 \(\end{matrix}\) \(\right\)]$, find each product, if possible. See Examples 5–7. BA

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.

$\(\begin{cases}\)2x + 6y + 6z = 8 \\2x + 7y + 6z = 10 \\2x + 7y + 7z = 9\(\end{cases}\)\\\(\text{The inverse of }\]\begin{bmatrix}\)2 & 6 & 6 \\2 & 7 & 6 \\2 & 7 & 7\(\end{bmatrix}\[\text{ is }\]\begin{bmatrix}\[\frac{7}{2}\) & 0 & -3 \\-1 & 0 & 0 \\0 & -1 & 1\(\end{bmatrix}\]\text{.}\)$

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. 1.5

x + 3y = 5

2x + 4y = 3

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