Prove property of adjoint: $(\mathcal{A}^{-1})^*=(\mathcal{A}^*)^{-1}$.

Question

I'm trying to prove it like any other property of adjoint. So, I need to prove following equality: $(\mathcal{A}^{-1}x, y)=(x, \mathcal({A}^{-1})^*y)$.

I know it's very basic, but how to prove this equality?

Or maybe this property of adjoint is proved in different way?

Thanks.

Answer 1

Hint based on the uniqueness of inverse of a matrix, assuming we work on an inner product linear space $\;V\;$ and, of course, on $\;\langle Au,v\rangle=\langle u, A^*v\rangle\;$ and $\;(A^*)^*=A\;$:

$$\forall\,u,v\in V:\;\langle\, A^*(A^{-1})^*u,v\,\rangle=\langle\,(A^{-1})^*u,Av\,\rangle=\langle\,u,A^{-1}Av\,\rangle=\langle u,v\rangle\implies A^*(A^{-1})^*=I$$

so...