If $A$ and $B$ are square matrices of the same order,then which of the following properties holds true?

If $A = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix}$,then prove that $A^n = \begin{bmatrix} 1+2n & -4n \\ n & 1-2n \end{bmatrix}$,where $n$ is any positive integer.

The number of all possible matrices of order $3 \times 3$ with each entry $0$ or $1$ is:

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

If $A = \begin{bmatrix} \sin \alpha & \cos \alpha \\ -\cos \alpha & \sin \alpha \end{bmatrix}$,then verify that $A^{\prime} A = I$.

If $A = \begin{bmatrix} 1 & 2 & x \\ 3 & -1 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} y \\ x \\ 1 \end{bmatrix}$ are such that $AB = \begin{bmatrix} 6 \\ 8 \end{bmatrix}$,then: