I need to prove/disprove that if A^2 is orthogonal matrix, than A is also orthogonal.

I think the statement is true, because if A^2 orthogonal, then A^tA^t = A^-1A^-1, hence AAA^tA^t = AAA^-1A^-1 = AIA^-1 = AA^-1 = I, hence A is orthogonal.

Is my proof right?