Math Help - transpose and symmetry

1. transpose and symmetry

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...)

2. $A\ =\ A^\mathrm TA$
$\implies\ A^\mathrm T\ =\ (A^\mathrm TA)^\mathrm T\ =\ A^\mathrm T(A^\mathrm T)^\mathrm T\ =\ A^\mathrm TA\ =\ A$
Hence $A$ is symmetric and $A=A^\mathrm TA=AA=A^2.$