Question

[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
\[
\operatorname{rank}(B C)+\operatorname{rank}(A B) \leq \operatorname{rank}(A B C)+\operatorname{rank}(B)
\]