Prove that $(A^\dagger)^\ast= (A^\ast)^\dagger $

Question

Let $A$ be an $m\times n$ matrix with coefficients in $\mathbb{C}$. Prove that $(A^\dagger)^\ast= (A^\ast)^\dagger $ where $A^\dagger$ is the Moore-Penrose Pseudoinverse of $A$.

My attempt:

\begin{align} (A^\dagger)^\ast &= ((A^\ast A)^{-1}A^\ast)^\ast\newline&= A^{\ast\ast}((A^\ast A)^{-1})^\ast \newline &= A(A^\ast A)^{-1} \end{align}

Then,

\begin{align} (A^\ast)^\dagger &= (A^{\ast\ast}A^\ast)^{-1}A^{\ast\ast}\newline&= (AA^\ast)^{-1}A \end{align}

Now I'm stuck, what am I doing wrong?

Answer 1

$$\begin{gather*}AA^*A=AA^*A\\ (AA^*)^{-1}AA^*A(A^*A)^{-1}=(AA^*)^{-1}AA^*A(A^*A)^{-1}\\ A(A^*A)^{-1} = (AA^*)^{-1}A\\ (A^{\dagger})^*=(A^*)^{\dagger} \end{gather*}$$