1. (10 pts) Let $A \in \mathsf{M}{m \times n}(\mathbb{R})$ and $B \in \mathsf{M}{n \times r}(\mathbb{R})$. If $AB = 0$, prove that
$$\operatorname{rank}(A) + \operatorname{rank}(B) \leq n.$$
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1. (10 pts) Let $A \in \mathsf{M}{m \times n}(\mathbb{R})$ and $B \in \mathsf{M}{n \times r}(\mathbb{R})$. If $AB = 0$, prove that
$$\operatorname{rank}(A) + \operatorname{rank}(B) \leq n.$$