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 $………$

If $A$ is a square matrix of order $3$ with $|A| = 2$, then the value of $|(A -A^T)^5| + |(A^T -A)^3|$ is-

If $A = \left[ {\begin{array}{*{20}{c}}1&{ – 2}&1\\2&1&3\end{array}} \right]$ and $B = \left[ {\begin{array}{*{20}{c}}2&1\\3&2\\1&1\end{array}} \right]$, then ${(AB)^T} = $

Let $\quad P_1=I=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right], \quad P_2=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{array}\right], \quad P_3=\left[\begin{array}{lll}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right], \quad P_4=\left[\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{array}\right], \quad P_5=\left[\begin{array}{lll}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{array}\right]$,

$P_6=\left[\begin{array}{lll}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{array}\right]$ and $X=\sum_{k=1}^6 P_k\left[\begin{array}{lll}2 & 1 & 3 \\ 1 & 0 & 2 \\ 3 & 2 & 1\end{array}\right] P_k^{\top}$

where $P _{ K }^{ T }$ denotes the transpose of the matrix $P _{ K }$. Then which of the following options is/are correct?

$(1)$ $X -30 I$ is an invertible matrix

$(2)$ The sum of diagonal entries of $X$ is 18

$(3)$ If $X \left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]=\alpha\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$, then $\alpha=30$

$(4)$ $X$ is a symmetric matrix

If $A = \left[ {\begin{array}{*{20}{c}}
1&1\\
0&1
\end{array}} \right]$ and $B = \left[ {\begin{array}{*{20}{c}}
{\frac{{\sqrt 3 }}{2}}&{\frac{1}{2}}\\
{\frac{{ – 1}}{2}}&{\frac{{\sqrt 3 }}{2}}
\end{array}} \right]$ , then $(BB^TA)^5$ is equal to