Does a symmetric matrix $A^2$ imply a symmetric $A$? Does a symmetric matrix $A^2$  imply a symmetric $A$? 
Any help would be much appreciated. 

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.
