最多的可交换的n阶矩阵

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REU12的第56题:

(Miklós Abért) Let $A_{1}, \ldots, A_{m}, B_{1}, \ldots, B_{m} \in M_{n}(F)$ be $n \times n$ matrices. Suppose that $A_{i} B_{j}=B_{j} A_{i}$ if and only if $i \neq j$. Prove: $m \leq n^{2}$.