Show that if $A \in M_n$ is similar to a unitary matrix, then $A^{-1}$ is similar to $A^*$

Question

I am trying to show that if $A \in M_n$ is similar to a unitary matrix, then $A^{-1}$ is similar to $A^*$. I know that $A = PUP^{-1}$ where U is a unitary matrix, and that $UU^* = I$, but I am not quite sure how to start this proof. Any input would be appreciated!

Answer 1

I think you mean that $A^{-1}$ is similar to $U^\ast$, if $A$ is similar to $U$. Recall that $(AB)^{-1} = B^{-1} A^{-1}$ is true for invertible matrices $A, B$. From $A = PUP^{-1}$ it now follows that

$$A^{-1} = (PUP^{-1})^{-1} = PU^{-1}P^{-1} = PU^\ast P^{-1},$$

hence $A^{-1}$ and $U^\ast$ are similar (note that the last equality follows from $UU^\ast = I$).