If $A$ and $B$ are square matrices of the same order,then which of the following properties holds true for the transpose of their product?

If $A = \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix}$,then $A' = $ . . . . . . .

If $A = \begin{bmatrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{bmatrix}$,then $A A^T$ is a

$P$ 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 $\operatorname{Tr}(P)=$

If $A = \begin{bmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{bmatrix}$,then $A + A^{\prime} = I$,if the value of $\alpha$ is

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 $AB = BA$.