If $A = \begin{bmatrix} 2 & -2 \\ -2 & 2 \end{bmatrix}$,then $A^n = 2^k A$,where $k = $

$A$ is a $2 \times 2$ matrix such that $A \begin{bmatrix} 1 \\ -1 \end{bmatrix} = \begin{bmatrix} -1 \\ 2 \end{bmatrix}$ and $A^2 \begin{bmatrix} 1 \\ -1 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$. The sum of the elements of $A$ is:

If $A=\left[\begin{array}{rrr}0 & 6 & 7 \\ -6 & 0 & 8 \\ 7 & -8 & 0\end{array}\right]$,$B=\left[\begin{array}{lll}0 & 1 & 1 \\ 1 & 0 & 2 \\ 1 & 2 & 0\end{array}\right]$,and $C=\left[\begin{array}{c}2 \\ -2 \\ 3\end{array}\right]$. Calculate $AC$,$BC$,and $(A + B)C$. Also,verify that $(A+B)C = AC + BC$.

$A$ and $B$ are two non-square matrices. If $P = A + B$, $Q = A^T B$, and $R = A B^T$, then the matrices whose order is equal to the order of $A$ are

Compute the following: $\begin{bmatrix} -1 & 4 & -6 \\ 8 & 5 & 16 \\ 2 & 8 & 5 \end{bmatrix} + \begin{bmatrix} 12 & 7 & 6 \\ 8 & 0 & 5 \\ 3 & 2 & 4 \end{bmatrix}$

If $A = \begin{bmatrix} 1 & -2 \\ 4 & 5 \end{bmatrix}$ and $f(t) = t^2 - 3t + 7$, then $f(A) + \begin{bmatrix} 3 & 6 \\ -12 & -9 \end{bmatrix}$ is equal to