TITLE: Why are eigenvalues of nilpotent matrices equal to zero?
QUESTION [21 upvotes]: If $A$ is a $ \displaystyle  10 \times 10 $ matrix such that $A^{3} = 0$ but $A^{2}  \neq 0$ (so A is nilpotent) then I know that $A$ is not invertible, but why does at least one eigenvalue of $A$ have to be equal to zero? How would one show that all eigenvalues of $A$ are equal to zero?

REPLY [9 votes]: Alternatively, do the contrapositive. If $A$ has a non-zero eigenvalue, $\lambda,$ then $A^{k} \neq 0 $ for all $k.$ 
Proof:
there exists $v \neq 0,$ such that 
$$
Av = \lambda v,
$$
so
$$
A^{k} v = \lambda^{k} v \neq 0.
$$
Done.