Let $A,B,C$ be complex-valued $n \times n$ matrices, where $\text{rank}(B) = n$. I know that for generic $A,B,C$ there is no relationship between $\text{rank}{(ABC)}$ and $\text{rank}(AC)$ like the similar $\text{rank}(AB) \leq \min(\text{rank}(A),\text{rank}(B) ), $ however if $B$ has full rank is it true that $$\text{rank}(ABC) = \text{rank}(AC)$$ or at least that $$\text{rank}(ABC) \leq \text{rank}(AC)?$$

  • $\begingroup$ No. Think of $A,B,C$ as linear maps. What happens e.g. if $C$ maps everything to $\ker(A)$? $\endgroup$ – amsmath Apr 19 at 18:10

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