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

linear-algebra matrices examples-counterexamples nilpotence

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


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

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

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