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

If $A$ is a symmetric matrix and $B$ is a skew-symmetric matrix such that $A + B = \begin{bmatrix} 2 & 3 \\ 5 & -1 \end{bmatrix}$,then $AB$ is equal to

If a square matrix $A$ is such that $\left(A^T-\frac{1}{2} I\right)\left(A-\frac{1}{2} I\right) = \left(A^T+\frac{1}{2} I\right)\left(A+\frac{1}{2} I\right) = I$,where $I$ is a unit matrix,then $A$ is

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

Let $A, B, C$ be $3 \times 3$ matrices such that $A$ is symmetric and $B$ and $C$ are skew-symmetric. Consider the statements:
$(S1): A^{13} B^{26} - B^{26} A^{13}$ is symmetric
$(S2): A^{26} C^{13} - C^{13} A^{26}$ is symmetric
Then,

$P$ is a $3 \times 3$ square matrix and $\operatorname{Tr}(P) \neq 0$. If $\operatorname{Tr}(P-P^{T})+\operatorname{Tr}(P+P^{T})+\frac{\operatorname{Tr}(P)}{\operatorname{Tr}(P^T)}+\operatorname{Tr}(P) \times \operatorname{Tr}(P^{T})=0$, then $\operatorname{Tr}(P)=$