The matrix $\begin{bmatrix} 0 & 5 & -7 \\ -5 & 0 & 11 \\ 7 & -11 & 0 \end{bmatrix}$ is known as:

$P$ is an orthogonal matrix and $A$ is a periodic matrix with period $4$,and $Q = PAP^T$. Then $X = P^TQ^{2005}P$ will be equal to

If for the matrix $A = \begin{bmatrix} 1 & -\alpha \\ \alpha & \beta \end{bmatrix}$,$AA^{T} = I_{2}$,then the value of $\alpha^{4} + \beta^{4}$ is ....... .

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

Let $X$ and $Y$ be two arbitrary,$3 \times 3$,non-zero,skew-symmetric matrices and $Z$ be an arbitrary $3 \times 3$,non-zero,symmetric matrix. Then which of the following matrices is (are) skew-symmetric?
$(A) Y^3 Z^4 - Z^4 Y^3$
$(B) X^{44} + Y^{44}$
$(C) X^4 Z^3 - Z^3 X^4$
$(D) X^{23} + Y^{23}$

Express the matrix $A = \left[\begin{array}{cc}1 & 5 \\ -1 & 2\end{array}\right]$ as the sum of a symmetric and a skew-symmetric matrix.