If $A$ and $B$ are symmetric matrices of the same order such that $AB+BA=X$ and $AB-BA=Y$,then $(XY)^{T}=$

Express the matrix $B=\left[\begin{array}{rrr}2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3\end{array}\right]$ as the sum of a symmetric and a skew symmetric matrix.

If $A$ is a square matrix and $A + A^T$ is a symmetric matrix,then $A - A^T$ is:

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?
$(A) Y^3 Z^4 - Z^4 Y^3$
$(B) X^{44} + Y^{44}$
$(C) X^4 Z^3 - Z^3 X^4$
$(D) X^{23} + Y^{23}$

Find $\frac{1}{2}(A+A^{\prime})$ and $\frac{1}{2}(A-A^{\prime}),$ when $A=\left[\begin{array}{ccc}0 & a & b \\ -a & 0 & c \\ -b & -c & 0\end{array}\right].$

If $ A = \begin{bmatrix} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{bmatrix} $,then $ A A^{\prime} = $