If A is an m x n matrix with rank(A) = n, then $A^{+} = (A^{T}A)^{-1}A^{T}$.
I already proved the first two of the Moore-Penrose equations. The second two are to verify:
3) $(AA^{+})^{*} = AA^{+}$
4) $(A^{+}A)^{*} = A^{+}A$
I've attempted substituting for $A^{+}$ on one side and moving different matrices around trying to derive the opposite side, but I'm lost outside of those first two moves.
The second equation is really easy to verify: note that $A^+A = (A^\top A)^{-1}A^\top A = I$.
For the first equation, recall these identities \begin{align*} (AB)^\top &= B^\top A^\top \\ (A^{-1})^\top &= (A^\top)^{-1} \\ (A^\top)^\top &= A. \end{align*} Then, \begin{align*} (A(A^\top A)^{-1}A^\top)^\top &= (A^\top)^\top ((A^\top A)^{-1})^\top A^\top \\ &= A ((A^\top A)^\top)^{-1} A^\top \\ &= A (A^\top (A^\top)^\top)^{-1} A^\top \\ &= A (A^\top A)^{-1} A^\top. \end{align*}