Question

Write a square matrix of order $2$, which is both symmetric and skew-symmetric.

Answer 1

Find square matrix of order $2$, which is both symmetric and skew symmetric:

A matrix $A$ is said to be symmetric if $A^T=A$.

A matrix $A$ is said to be skew-symmetric if $A^T=-A$

Take $A=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$.

Then,

$\begin{array}{rcl}{A}^T& =& \left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\\ & \Rightarrow & A^T=A\\ A^T& =& \left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\\ & \Rightarrow & A^T=-\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\\ & \Rightarrow & A^T=-A\end{array}$

So, $A=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$ is both symmetric and skew-symmetric.

Hence, $A=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$ is a square matrix of order $2$ which is both symmetric and skew-symmetric.