[Math] Show that if $A$ is any square matrix such that $A^n = 0$ for some positive intiger $n$, then $A$ is not invertible. (answer check)

Question

Show that if $A$ is any square matrix such that $A^n = 0$ for some positive integer $n$, then $A$ is not invertible.

I'm not sure if my proof is good enough, or enough "work" as my teacher put it after my last test.

This is what I have:

$$A^n = 0$$

or $A$ to be invertible,
$$A^n A^{-n} = I$$

Then:

$$A^{n} \cdot A^{n} =0 \cdot A^{-n} $$

$$I \neq 0$$

this is not true, so $A$ is not invertible.

Answer 1

Hint:If $A$ is nilpotent operator, then its only eigenvalue is zero.