Explanation: If matrix $\mathrm{A}$ is symmetric
${\mathit{A}}^{\mathit{T}}=\mathit{A}$
If matrix $\mathrm{A}$ is skew-symmetric
${\mathit{A}}^{\mathit{T}}=-\mathit{A}$
Also, diagonal elements are zero.
Since, it is given that matrix $\mathrm{A}$ is both symmetric and skew-symmetric.
$\therefore \mathit{A}={\mathit{A}}^{\mathit{T}}=-\mathit{A}$
Which is only possible if $\mathit{A}$ is zero matrix.
$\mathit{A}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]={\mathit{A}}^{\mathit{T}}=-\mathit{A}$
Thus, if a matrix A is both symmetric and skew symmetric, then $\mathrm{A}$ is necessarily a zero matrix.