Does there exist a non-symmetric involutory matrix? 
Let $A$ be a real involutory matrix i.e. $$A^2 = I.$$

Is it necessarily symmetric? 
Any help will be highly appreciated. Thank you very much.

It is easy to construct 2-dimensional counterexamples:  $A = \begin{pmatrix}a & b \\ c & d\end{pmatrix}$ is involutory if $a+d=0$ and $a^2+bc=1$. In particular, $A = \begin{pmatrix}0 & b \\ 1/b & 0\end{pmatrix}$  with $b \ne 0, 1$ is a non-symmetric, involutory matrix.
