TITLE: If $A$ is a square matrix and $A^2 = 0$ then $A=0$. Is this true? If not, provide a counter-example.
QUESTION [6 upvotes]: This is a proof question and I am not sure how to prove it. It is obviously true if you start with $A = 0$ and square it. 
I was thinking:
If $ A^2 = 0 $
then
$ A A = 0 $
$ A A A^{-1} = 0 A^{-1}$
$I\,A = 0 $ 
but the zero matrix is not invertible and that it was not among the given conditions.
Where's a good place to start?

REPLY [4 votes]: Given two nonzero orthogonal vectors $u, v \in \mathbb{R}^n$. Let $A = vu^T$, then
$$
A^2 = vu^Tvu^T = 0
$$