In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. $\(\begin{cases}\) 5x + 12y + z = 10 \\ 2x + 5y + 2z = -1 \\ x + 2y - 3z = 5 \(\end{cases}\)$

Write the augmented matrix for each system of linear equations.

$\(\begin{cases}\) 2x + y + 2z = 2 \\ 3x - 5y - z = 4 \\ x - 2y - 3z = -6 \(\end{cases}\)$

a. Give the order of each matrix.

b. If $A=\left[a_{ij}\right]$, identify $a_{32}$ and $a_{23}$, or explain why identification is not possible.

$\(\begin{bmatrix}\) 4 & -7 & 5 \\ -6 & 8 & -1 \(\end{bmatrix}\)$

Perform each matrix row operation and write the new matrix.

$\(\begin{bmatrix}\) 1 & 2 & 2 & \(\vert\) & 2 \\ 0 & 1 & -1 & \(\vert\) & 2 \\ 0 & 5 & 4 & \(\vert\) & 1 \(\end{bmatrix}\) -5R_2 + R_3$

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

$A = \(\begin{bmatrix}\) 4 & -3 \\ -5 & 4 \(\end{bmatrix}\) , B = \(\begin{bmatrix}\) 4 & 3 \\ 5 & 4 \(\end{bmatrix}\)$

Evaluate each determinant in Exercises 1–10.

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

a. Give the order of each matrix.

b. If A = [aᵢⱼ] , identify a₃₂ and a₂₃, or explain why identification is not possible.

$\(\begin{bmatrix}\) 1 & -5 & \(\pi\) & e \\ 0 & 7 & -6 & -\(\pi\) \\ -2 & \(\frac{1}{2}\) & 11 & -\(\frac{1}{5}\) \(\end{bmatrix}\)$