Prove similarity of matrix $A^{-1}$ to matrix $A^{*}$ which is Hermitian adjoint Let $A \in \mathcal M_{n}(\Bbb C)$ and $A$ is similar to  unitary matrix.
Prove that $A^{-1}$ is similiar to $A^{*}$, where $A^{*}$ is Hermitian adjoint.
$A = C^{-1}UC$, where $U$ is unitary matrix
So $A^{-1} = (C^{-1}UC)^{-1} = C^{-1}U^{-1}C \iff U^{-1}=CA^{-1}C^{-1}$
$A^{*} = \overline{(A^{T})} = \overline{(C^{-1}UC)^T} = \overline{C^T}\cdot\overline{U^T}\cdot\overline{(C^{-1})^T}$, but U is unitary so
$\overline{U^T} = U^{-1}$
Hence $A^{*} = \overline{C^T}\cdot CA^{-1}C^{-1} \cdot\overline{(C^{-1})^T}$
What can I do next? Moreover, is it at least correct?

What you have is correct. You can now say that
$$
A^*= (C^*C)A^{-1}(C^*C)^{-1}
$$
By definition, this means $A^*$ is similar to $A^{-1}$.
