The matrix $A = \begin{bmatrix} 1/\sqrt{2} & 1/\sqrt{2} \\ -1/\sqrt{2} & -1/\sqrt{2} \end{bmatrix}$ is

$A, B, C, D$ are square matrices such that $A+B$ is symmetric, $A-B$ is skew-symmetric and $D$ is the transpose of $C$. If $A=\left[\begin{array}{ccc}-1 & 2 & 3 \\ 4 & 3 & -2 \\ 3 & -4 & 5\end{array}\right]$ and $C=\left[\begin{array}{ccc}0 & 1 & -2 \\ 2 & -1 & 0 \\ 0 & 2 & 1\end{array}\right]$, then the matrix $B+D=$

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

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

If the matrix $\begin{bmatrix} a & 2 & -3 \\ b & 0 & 4 \\ c & -4 & 0 \end{bmatrix}$ is a skew-symmetric matrix,then $a+b+c=$

If $P$ is a $3 \times 3$ matrix such that $P^{\top}=2 P+I$,where $P^{\top}$ is the transpose of $P$ and $I$ is the $3 \times 3$ identity matrix,then there exists a column matrix $X=\left[\begin{array}{l}x \\ y \\ z\end{array}\right] \neq\left[\begin{array}{l}0 \\ 0 \\ 0\end{array}\right]$ such that