If $A$ is a $3 \times 4$ matrix and $B$ is a matrix such that $A^{\prime}B$ and $BA^{\prime}$ are both defined,then $B$ is of the type:

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

Let $A$ and $B$ be two symmetric matrices of the same order. Then, the matrix $AB - BA$ is

Let $A$ be a square matrix such that $a_{ij} \in \{-1, 0, 1\}$ for all $i, j$ and it has exactly one non-zero entry in each row as well as in each column. Then:

If $A$ is a nilpotent matrix of index $2$,then $A(I_2+A)^{51}$ is equal to (where $I_2$ is the identity matrix of order $2$)

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: