The total number of matrices $A = \left[ {\begin{array}{*{20}{c}}
0&{2x}&{2x}\\
{2y}&y&{ – y}\\
1&{ – 1}&1
\end{array}} \right];\,\left( {x,y \in R,\,x \ne y} \right)$ for which ${A^T}A = 3{I_3}$

If $A=\left(\begin{array}{cc}\frac{1}{\sqrt{5}} & \frac{2}{\sqrt{5}} \\ \frac{-2}{\sqrt{5}} & \frac{1}{\sqrt{5}}\end{array}\right), B=\left(\begin{array}{ll}1 & 0 \\ i & 1\end{array}\right), i=\sqrt{-1}$, and

$\mathrm{Q}=\mathrm{A}^{\mathrm{T}} \mathrm{BA}$, then the inverse of the matrix $\mathrm{A} \mathrm{Q}^{2021} \mathrm{~A}^{\mathrm{T}}$ 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 $\left(\mathrm{A}^{\prime}\right)^{\prime}=\mathrm{A}$.

If $ A$  is a square matrix, then which of the following matrices is not symmetric

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