If characteristic polynomial of $A$ is $(x-1)^n$, then show $A$ and $A^{-1}$ are similar

Question

I have a problem about matrix similarity:

Let $A$ be an $n\times n$ complex matrix whose characteristic polynomial is $(x-1)^n$. I want to show that $A$ and $A^{-1}$ are similar matrices.

I know that both $A$ and $A^{-1}$ have $1$ as their only eigenvalue, with algebraic multiplicity $n$. But I don't know how to use this fact to prove their similarity. Can anyone give me a hint or a solution?

Answer 1

Let $J$ be the Jordan Canonical form of $A$. Then $J=P^{-1}AP$ for some invertible matrix. So $J^{-1}=P^{-1}A^{-1}P$. Now observe that $J$ and $J^{-1}$ are similar. So $A$ and $A^{-1}$ are similar.