Find the number of $3 \times 3$ matrices having all elements either $2$ or $9$.

If $A = \begin{bmatrix} 2 & -1 \\ 0 & 2 \end{bmatrix}$ and $A^2 + xA + yI_2 = O_2$, where $I_2$ and $O_2$ are the identity matrix and null matrix of order $2$ respectively, then:

The number of matrices with order $3 \times 2$ whose each entry is $1$ or $2$ is . . . . . . .

If $A = \begin{bmatrix} 1 & 2 \\ 4 & 2 \end{bmatrix}$,then show that $|2A| = 4|A|$.

If $A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$,then ${A^n} = $

If $A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$,then $AB = $