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} = $

If both $\left( {A – \frac{I}{2}} \right)$ and ${A + \frac{I}{2}}$ are orthogonal matrices, then  

Let $X$ and $Y$ be two arbitrary, $3 \times 3$, non-zero, skew-symmetric matrices and $Z$ be an arbitrary $3 \times 3$, nonzero, symmetric matrix. Then which of the following matrices is (are) skew symmetric?

$(A)$ $Y^3 Z^4-Z^4 Y^3$ $(B)$ $X ^{44}+ Y ^{44}$

$(C)$ $X ^4 Z ^3- Z ^3 X ^4$ $(B)$ $X ^{23}+ Y ^{23}$

How many $3 \times 3$ matrices $\mathrm{M}$ with entries from $\{0,1,2\}$ are there, for which the sum of the diagonal entries of $M^T M$ is $5$ ?

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