Prove that $T$ has at most two distinct eigenvalues

Question

I need help with a problem in Axler's Linear Algebra text.
Any hint would be great.

Let $V$ be a vector space over $\mathbb{C}$ of dimension $n$, and $T: V\to V.$ If $$\dim \ker(T^{n-2}) \neq \dim \ker(T^{n-1}),$$ then prove that $T$ has at most two distinct eigenvalues.

Answer 1

Hint: Use the information given to conclude that the the Jordan canonical form of $T$ has a Jordan block associated with $0$ of size at least $n-1$.