# Relationship between $\text{rank} ABC$ and $\text{rank} AC$ where $B$ has full rank

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 $$\text{rank}(ABC) \leq \text{rank}(AC)?$$

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