If matrix $A$ is unitary and $B^2 = A$ then $B$ is also unitary

Question

I need to prove or give a counterexample:

If matrix $A$ is unitary and $B^2 = A$ then $B$ is also unitary

I think the statement is true since the unitary matrix A can only be Identity matrix I or negative identity matrix $-I$; and $B=A^2$ is an identity matrix which makes sure it is unitary.

Answer 1

HINT

You are incorrect. Both $I$ and $-I$ are, of course, unitary, but there are many such matrices, e.g. $$ A = \begin{bmatrix} \sqrt{2}/2 & -\sqrt{2}/2 \\ \sqrt{2}/2 & \sqrt{2}/2 \end{bmatrix} $$

But think: matrix $M$ is unitary if and only if $M^T M = I$. What could you say above $(M^2)^T (M^2)$ in such circumstances?

Your question asks to go the other way. If $M^TM = I$ and $A^2 = M$, can we say that $A^TA = I$ as well?