Does a symmetric matrix $A^2$ imply a symmetric $A$?

Question

Does a symmetric matrix $A^2$ imply a symmetric $A$?

Any help would be much appreciated.

Answer 1

No. $$ \left( \begin{array}{cc} 0 & 1 \\ 2 & 0 \end{array} \right) \left( \begin{array}{cc} 0 & 1 \\ 2 & 0 \end{array} \right) = \left( \begin{array}{cc} 2 & 0 \\ 0 & 2 \end{array} \right) $$ which is symmetric.

Answer 2

HINT:

No, for instance take $A$ antisymmetric.

Answer 3

Hint: No, for instance some nilpotent matrices.