How to prove that the inverse of a conjugate matrix is equal to the conjugate of an inverse of the same matrix?

Question

I am a little confused as to proving that $(C^*)^{-1} = (C^{-1})^*$ where $C$ is an invertible matrix which is complex.

Initially, I thought that it would have something to do with the identity matrix where $CC^{-1}=C^{-1}$. $C = I$ but don't seem to be getting anywhere with that.

Thank you!

Answer 1

You also need to use $(AB)^\ast =B^\ast A^\ast$ with $A=C,\,B=C^{-1}$. Both sides are $I$; the right-hand side must therefore be $(C^\ast)^{-1}C^\ast$.

Answer 2

*: conjugation (not Hermitian) \begin{equation} I=AA^{-1}=A^*(A^{-1})^* \end{equation} since $I=I^*$.

Also, \begin{equation} I=A^*(A^*)^{-1}. \end{equation}

Therefore \begin{equation} A^*(A^*)^{-1}=A^*(A^{-1})^*. \end{equation}

Multiplying from left by $(A^*)^{-1}$ you get \begin{equation} (A^*)^{-1}=(A^{-1})^*. \end{equation}