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

Let $M$ be a $3 \times 3$ matrix satisfying $M\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} -1 \\ 2 \\ 3 \end{bmatrix}$,$M\begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \\ -1 \end{bmatrix}$,and $M\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 12 \end{bmatrix}$. Then the sum of the diagonal entries of $M$ is

The order of matrix $A$ is $m \times n$ and for matrix $B$,if $AB^{\prime}$ and $B^{\prime}A$ are defined,then the order of matrix $B$ is . . . . . . .

Assume $X, Y, Z, W$ and $P$ are the matrices of order $2 \times n, 3 \times k, 2 \times p, n \times 3$ and $p \times k$ respectively. The restriction on $n, k$ and $p$ so that $PY + WY$ will be defined are:

Let $T$ and $U$ be the set of all orthogonal matrices of order $3$ over $\mathbb{R}$ and the set of all non-singular matrices of order $3$ over $\mathbb{R}$ respectively. Let $A = \{-1, 0, 1\}$. Then:

If $P = \begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{bmatrix}$, then $P^5$ is equal to