If $A$ and $B$ are symmetric matrices of the same order, then show that $AB$ is symmetric if and only if $A$ and $B$ commute, that is $AB = BA$.

Let $A$ be a symmetric matrix such that $|A|=2$ and $\left[\begin{array}{ll}2 & 1 \\ 3 & \frac{3}{2}\end{array}\right] A =\left[\begin{array}{ll}1 & 2 \\ \alpha & \beta\end{array}\right]$. If the sum of the diagonal elements of $A$ is s, then $\frac{\beta s}{\alpha^2}$ is equal to $……….$.

If the matrix $\mathrm{A}$ is both symmetric and skew symmetric, then

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 $X$ 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?

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

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