Prove if (transpose of A)*A = A, then A is symmetric and A = A^2:

I think I understand it, but Im not sure if I can put it into proper terms

Since the transpose of A = A (if symmetric), then if you multiply both sides of the equation, you get (Transpose of A)*A = A^2

so therefore A must = A^2 (using the original statement...if (transpose of A)*A = A...)

Any clarification please?