7. Systems of Equations & Matrices

In Exercises 23–30, use expansion by minors to evaluate each determinant. $\(\begin{vmatrix}\)1 & 1 & 1 \\2 & 2 & 2 \\-3 & 4 & -5\(\end{vmatrix}\)$

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.

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\(\end{cases}\)$

Evaluate the determinant of the matrix.

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

$y=-3x+4$

$-2x=7y-9$

Evaluate the determinant of the matrix.

Solve the system of equations using Cramer's Rule.

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

$x+y+z=3$

$5x+y+2z=5$

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

$y-9x=-3$

$-3x=4y-1$

What is the value of ?

Evaluate each determinant in Exercises 1–10.

$\(\begin{vmatrix}\)5 & 7 \\2 & 3\(\end{vmatrix}\)$

What expression in x represents ?

Evaluate each determinant in Exercises 1–10.

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

Evaluate each determinant in Exercises 1–10.

$\(\begin{vmatrix}\)-7 & 14 \\2 & -4\(\end{vmatrix}\)$