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

If $A$ and $B$ are symmetric matrices of the same order,then $AB - BA$ is . . . . . . .

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 $(A')' = A$.

Let $A + 2B = \begin{bmatrix} 1 & 2 & 0 \\ 6 & -3 & 3 \\ -5 & 3 & 1 \end{bmatrix}$ and $2A - B = \begin{bmatrix} 2 & -1 & 5 \\ 2 & -1 & 6 \\ 0 & 1 & 2 \end{bmatrix}$. Then $Tr(A) - Tr(B)$ has the value equal to:

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

For a square matrix $A$,if $A = B + \frac{C}{2}$,where $B$ is a skew-symmetric matrix and $C$ is a symmetric matrix,then $C = $ . . . . . . .