If $A$ is a square matrix and $A^2 = 0$ then $A=0$. Is this true? If not, provide a counter-example.
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?
$\endgroup$ 23 Answers
$\begingroup$HINT: Consider $A = \begin{bmatrix}0 & 1\\0 & 0\end{bmatrix}$
$\endgroup$ $\begingroup$Given two nonzero orthogonal vectors $u, v \in \mathbb{R}^n$. Let $A = vu^T$, then
$$ A^2 = vu^Tvu^T = 0 $$
$\endgroup$ $\begingroup$consider $$A=\begin{bmatrix} 2 & 1\\ -4 & -2 \end{bmatrix}$$
$\endgroup$ 0
