If $A = \begin{bmatrix} 5 & 2x+3 \\ x-2 & x+1 \end{bmatrix}$ is a symmetric matrix,then $x$ is 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 ....... .

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,

If $A = \begin{bmatrix} 1 & -2 \\ 5 & 3 \end{bmatrix}$,then $A + A^T$ equals:

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.

For any square matrix $A$,$AA^T$ is a