$A$ and $B$ are two given matrices such that the order of $A$ is $3×4$ , if $A’ B$ and $BA’$ are both defined then

If $A$  and $ B$ are square matrices of the same order, then

Let $A=\left[\begin{array}{cc}0 & -2 \\ 2 & 0\end{array}\right]$. If $M$ and $N$ are two matrices given by $M =\sum \limits_{ k =1}^{10} A ^{2 k }$ and $N =\sum \limits_{ k =1}^{10} A ^{2 k -1}$ then $MN ^{2}$ is

Let $P =\left[\begin{array}{ccc}3 & -1 & -2 \\ 2 & 0 & \alpha \\ 3 & -5 & 0\end{array}\right]$ where $\alpha \in R .$ Suppose $Q =\left[ q _{ ij }\right]$ is a matrix satisfying $PQ = kI _{3}$ for some non-zero $k \in R .$ If $q_{23}=-\frac{k}{8}$ and $|Q|=\frac{k^{2}}{2}$, then $\alpha^{2}+ k ^{2}$ is equal to………..

If $A=\left[\begin{array}{lll}3 & \sqrt{3} & 2 \\ 4 & 2 & 0\end{array}\right]$ and $B=\left[\begin{array}{rrr}2 & -1 & 2 \\ 1 & 2 & 4\end{array}\right],$ verify that $(A+B)^{\prime}=A^{\prime}+B^{\prime}$.