If $A$ is a square matrix and $A^2 = 0$ then $A=0$. Is this true? If not, provide a counter-example.

Question

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?

Answer 1

HINT: Consider $A = \begin{bmatrix}0 & 1\\0 & 0\end{bmatrix}$

Answer 2

consider $$A=\begin{bmatrix} 2 & 1\\ -4 & -2 \end{bmatrix}$$

Answer 3

Given two nonzero orthogonal vectors $u, v \in \mathbb{R}^n$. Let $A = vu^T$, then

$$ A^2 = vu^Tvu^T = 0 $$