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Is the result $|\text{rank(AB)}-\text{rank(BA)}| \le \text{min}(\text{rank}(A),\text{rank}(B))$ true for all $n*n$ matrices $A,B$. I only know the result that $\text{rank(AB)} \le \text{min(rank(A),rank(B))}$

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Yes. Your desired inequality is a consequence of the one you mentioned. In other words, it follows from the fact that the ranks of both $AB$ and $BA$ are non-negative integers at most equal to that minimum.

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  • $\begingroup$ Yes, it's really just a question about integer arithmetic! $\endgroup$ – Robert Lewis Sep 14 '19 at 16:22

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