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[Frobenius] Let $A, B$, and $C$ be matrices so that the products $A B$ and $B C$ are defined. Use the obvious
$\operatorname{dim}\left(\left.\operatorname{ker} A\right|_{\operatorname{Im}} B C\right)=\operatorname{dim} \operatorname{Im} B C-\operatorname{dim} \operatorname{Im} A B C$
and the previous part to show that
$\operatorname{rank}(B C)+\operatorname{rank}(A B) \leq \operatorname{rank}(A B C)+\operatorname{rank}(B)$
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