Question

If a matrix $A$ is both symmetric and skew symmetric, then?

Answer 1

Find the matrix $A$

It is given that a matrix $A$ is both symmetric and skew symmetric:

$$\begin{gathered} {\therefore A^T}^{}=A \\ {A}^T=-A \end{gathered}$$

Comparing both the equation we get

$$\begin{gathered} A=-A \\ \Rightarrow A+A=0 \\ \Rightarrow 2A=0 \end{gathered}$$

This means that $A$ is a zero matrix.

Hence, when $A$ is both symmetric and skew symmetric, then $A$ is a zero matrix.