If $A = \begin{bmatrix} 1 & 0 \\ -1 & 7 \end{bmatrix}$,$I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ and $A^{2} = 8A + kI$,then the value of $k$ is

If $A = \begin{bmatrix} 1 & a \\ 0 & 1 \end{bmatrix}$,then $A^n$ (where $n \in N$) equals

Let $A = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}$. Then the number of $3 \times 3$ matrices $B$ with entries from the set $\{1, 2, 3, 4, 5\}$ and satisfying $AB = BA$ is $....$

If $A = \begin{bmatrix} 8 & 0 \\ 4 & -2 \\ 3 & 6 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & -2 \\ 4 & 2 \\ -5 & 1 \end{bmatrix}$,then find the matrix $X$ such that $2A + 3X = 5B$.

If $A = \begin{bmatrix} ab & b^2 \\ -a^2 & -ab \end{bmatrix}$ and $A^n = O$,then the minimum value of $n$ is

For the matrix $A = \begin{bmatrix} 3 & 2 \\ 1 & 1 \end{bmatrix}$,find the values of $a$ and $b$ such that $A^{2} + aA + bI = 0$.