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 $………$

Let $\quad P=\left[\begin{array}{cc}\frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2}\end{array}\right], A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]$ and $Q=P Q P^{ T }$. If $P ^{ T } Q ^{2007} P =\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]$, then $2 a+b-3 c-4 d$ equal to $……………….$.

Let $A$ is a symmetric and $ \,B$ is a skew symmetric matrix, such that  $A – B = \left[ {\begin{array}{*{20}{c}}
  1&2 \\ 
  3&4 
\end{array}} \right]$, then $|A|$ is

Let three matrices $A =$$\left[ {\begin{array}{*{20}{c}}2&1\\4&1\end{array}} \right]$ ; $B =$$\left[ {\begin{array}{*{20}{c}}3&4\\2&3\end{array}} \right]$ and $C =$$\left[ {\begin{array}{*{20}{c}}3&{ – 4}\\{ – 2}&3\end{array}} \right]$ then $T_r(A) + t_r$ $\left( {\frac{{ABC}}{2}} \right)$ $+$ $t_r$$\left( {\frac{{A{{(BC)}^2}}}{4}} \right)$  $+$ $t_r$ $\left( {\frac{{A{{(BC)}^3}}}{8}} \right)$ $+$ ……. $+ \infty =$

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