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

In a skew-symmetric matrix,the diagonal elements are all

If $P$ and $Q$ are symmetric matrices of the same order,then $PQ - QP$ is

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 $A = \begin{bmatrix} 3 & x-1 \\ 2x+3 & x+2 \end{bmatrix}$ is a symmetric matrix,then the value of $x$ is

$A=\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$ and $B=\begin{bmatrix} x & y \\ 1 & 2 \end{bmatrix}$ are two matrices such that $(A+B)(A-B)=A^2-B^2$. If $C=\begin{bmatrix} x & 2 \\ 1 & y \end{bmatrix}$,then $\operatorname{Trace}(C)=$