Matrix similar to its inverse

Question

I have this problem:

$A$ is an $n \times n$-matrix, its characteristic polynomial is $P(X)=(X-1)^n$. Prove that $A$ is similar to its inverse.

How do you solve it? I really don't know.

Answer 1

Hints. Call the underlying field $\mathbb{F}$. We will use the following fact:

If the characteristic polynomial of a matrix $B\in M_n(\mathbb{F})$ can be factored into linear factors over $\mathbb{F}$ (i.e. if $B$ has a complete set of eigenvalues in $\mathbb{F}$), $B$ is similar to its Jordan form over $\mathbb{F}$.

1. Show that $A$ is is similar to its Jordan form over $\mathbb{F}$. So, it suffices to prove the problem statement for a single Jordan block.
2. From now on, suppose $A$ is an $n\times n$ Jordan block whose characteristic polynomial is $(x-1)^n$. Show that $A$ is invertible.
3. For any positive integer $m$, prove that $(A-I)^m=0$ if and only if $(A^{-1}-I)^m=0$.
4. Hence show that the minimal polynomial of $A^{-1}$ is $(x-1)^n$ and that $A^{-1}$ is similar to $A$ over $\mathbb{F}$.