Question

Write a $2$ $\times$ $2$ matrix which is both symmetric and skew-symmetric.

Answer 1

For a symmetric matrix $A=A^T$

For a skew-symmetric matrix $-A=A^T$

So, if a matrix is both symmetric and skew-symmetric, then

$A=-A$

$\Rightarrow 2A=O$, a null matrix

$\Rightarrow A=O$, a null matrix

Hence, a $2\times 2$ matrix which is both symmetric and skew-symmetric is

$\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$