7. (10 points) $A = \begin{bmatrix} 0 & -2 & 1 \\ 4 & -2 & 1 \end{bmatrix}$, $B = \begin{bmatrix} 5 & -3 \\ -1 & 0 \\ 0 & 4 \end{bmatrix}$, $C = \begin{bmatrix} -2 & 1 \\ 3 & 0 \end{bmatrix}$. \\
Calculate $AB$, $A^T C$, and $AB - A^T C$ if these operations are valid. If some of them are not valid \\
explain why.\\
8. (10 points) If $A = \begin{bmatrix} 0 & 5 \\ 2 & 0 \end{bmatrix}$ and $B = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ find conditions on $a$, $b$, $c$, and $d$ such that \\
$AB - BA^T = O$ \\
9. (10 points) Let $A = \begin{bmatrix} 1 & -1 & 0 \\ 2 & 1 & 3 \\ -1 & 4 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & -1 & 0 \\ 0 & 3 & 3 \\ -1 & 4 & 1 \end{bmatrix}$ \\
(a) (6 points) Find an elementary matrix $E$ such that $EA = B$. \\
(b) (4 points) Evaluate $E^{-1}$.
