Let $M$ be any $3 \times 3$ matrix with entries from the set $\{0,1,2\}$. The maximum number of such matrices, for which the sum of diagonal elements of $M ^{ T } M$ is seven, is ………….

The matrix $A = \left[ {\begin{array}{*{20}{c}}{1/\sqrt 2 }&{1/\sqrt 2 }\\{ – 1/\sqrt 2 }&{ – 1/\sqrt 2 }\end{array}} \right]$ is

Let $A=\left[\begin{array}{cc}\frac{1}{\sqrt{10}} & \frac{3}{\sqrt{10}} \\ \frac{-3}{\sqrt{10}} & \frac{1}{\sqrt{10}}\end{array}\right]$ and $B =\left[\begin{array}{rr}1 & – i \\ 0 & 1\end{array}\right]$, where $i =\sqrt{-1}$. If $M = A ^{ T } BA$, then the inverse of the matrix $AM ^{2023} A ^{ T }$ is $………$

Express the following matrices as the sum of a symmetric and a skew symmetric matrix : $\left[\begin{array}{cc}1 & 5 \\ -1 & 2\end{array}\right]$

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 $(A+B)^{\prime}=A^{\prime}+B^{\prime}$.