# How to prove and interpret $\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B))$?

Let $$A$$ and $$B$$ be two matrices which can be multiplied. Then $$\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B)).$$

I proved $$\operatorname{rank}(AB) \leq \operatorname{rank}(B)$$ by interpreting $$AB$$ as a composition of linear maps, observing that $$\operatorname{ker}(B) \subseteq \operatorname{ker}(AB)$$ and using the kernel-image dimension formula. This also provides, in my opinion, a nice interpretation: if non-stable, under subsequent compositions the kernel can only get bigger, and the image can only get smaller, in a sort of loss of information.

How do you manage $$\operatorname{rank}(AB) \leq \operatorname{rank}(A)$$? Is there a nice interpretation like the previous one?