If $A$ is a matrix of order $m \times n$ and $B$ is a matrix such that $AB^{\prime}$ and $B^{\prime}A$ are both defined,then the order of the matrix $B$ is

If $A = \begin{bmatrix} 1 & 0 & 2 \\ 2 & 1 & 3 \\ 3 & 2 & 4 \end{bmatrix}$,then $A^2 - 5A + 6I =$

Let $I$ denote the $3 \times 3$ identity matrix and $P$ be a matrix obtained by rearranging the columns of $I$. Then,

Compute the indicated product: $\begin{bmatrix} a & b \\ -b & a \end{bmatrix} \begin{bmatrix} a & -b \\ b & a \end{bmatrix}$

The number of matrices of order $3 \times 3$,whose entries are either $0$ or $1$ and the sum of all the entries is a prime number,is:

In the matrix $A = \begin{bmatrix} 2 & 5 & 19 & -7 \\ 35 & -2 & \frac{5}{2} & 12 \\ \sqrt{3} & 1 & -5 & 17 \end{bmatrix}$,write:
$(i)$ The order of the matrix
$(ii)$ The number of elements
$(iii)$ The elements $a_{13}, a_{21}, a_{33}, a_{24}, a_{23}$