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Let $p$ be a prime and let $\mathbb{F}_p=\mathbb{Z} / p \mathbb{Z}$. Find all $p \times p$ matrices $A$ and $B$ over $\mathbb{F}_p$ such that $AB - BA = \mathbb{I}$.

Question: Can you do that in characheristic zero or for $n \times n$ matrices where $p$ does not divide $n$ ? Give a brief explanation.