Let $X$ and $Y$ be two arbitrary,$3 \times 3$,non-zero,skew-symmetric matrices and $Z$ be an arbitrary $3 \times 3$,non-zero,symmetric matrix. Then which of the following matrices is (are) skew-symmetric?
$(A) Y^3 Z^4 - Z^4 Y^3$
$(B) X^{44} + Y^{44}$
$(C) X^4 Z^3 - Z^3 X^4$
$(D) X^{23} + Y^{23}$

If $A = \begin{bmatrix} 3 & \sqrt{3} & 2 \\ 4 & 2 & 0 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & -1 & 2 \\ 1 & 2 & 4 \end{bmatrix}$,verify that $(A')' = A$.

If $A$ and $B$ are symmetric matrices,then $ABA$ is

For any square matrix $A$,$AA^T$ is a

If $A$ is a skew-symmetric matrix,then (given $n \in N$):
$1$. $A^{2n}$ is a skew-symmetric matrix.
$2$. $A^{2n+1}$ is a skew-symmetric matrix.

Let $A+2 B=\left[\begin{array}{ccc}1 & 2 & 0 \\ 6 & -3 & 3 \\ -5 & 3 & 1\end{array}\right]$ and $2 A - B =\left[\begin{array}{ccc}2 & -1 & 5 \\ 2 & -1 & 6 \\ 0 & 1 & 2\end{array}\right] .$ If $\operatorname{Tr}( A )$ denotes the sum of all diagonal elements of the matrix $A ,$ then $\operatorname{Tr}( A )-\operatorname{Tr}( B )$ has value equal to