If $\quad A=\left[\begin{array}{rrr}1 & 1 & -1 \\ 2 & 0 & 3 \\ 3 & -1 & 2\end{array}\right]$,  $B=\left[\begin{array}{rr}1 & 3 \\ 0 & 2 \\ -1 & 4\end{array}\right]$ and $C=\left[\begin{array}{rrrr}1 & 2 & 3 & -4 \\ 2 & 0 & -2 & 1\end{array}\right],$ find $A(B C),(A B) C$ and show that $(A B) C=A(B C)$.

Find $x,$ if $[x-5-1]\left[\begin{array}{lll}1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3\end{array}\right]\left[\begin{array}{l}x \\ 4 \\ 1\end{array}\right]=O$

$A$ and $B$ be $3 \times 3$ matrices such that $AB + A + B = 0$ , then

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 $P Y+W Y$ will be defined are :

Let $A=\left(\begin{array}{cc}1+ i & 1 \\ – i & 0\end{array}\right)$ where $i =\sqrt{-1}$ Then, the number of elements in the set $\left\{ n \in\{1,2, \ldots, 100\}: A ^{ n }= A \right\}$ is