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### Rank of matrix products is never greater than what [duplicate]

If I have a product of matrices of rank 1, the product is not going to have rank greater than 1. why?
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### Rank of product of two matrices [duplicate]

I want to show that $\text {rank} ( AB)\le \min(\text{rank} B, \text{rank} A)$ and when the equality occurs? Please help me with this problem by giving hints, solving, or suggesting a book. Thanks
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### What can be said about $\text{Rank}(C)?$ [duplicate]

Assume that $A\in\mathbb{R}^{n\times k}, \ B\in\mathbb{R}^{k\times m}$ and that $\text{Rank}(A)=r, \ \text{Rank}(B)=s.$ What can be said about $\text{rank}(C),$ where $C=AB?$ The only thing I can ...
### Let $p(t), q(t) ∈ \mathbb C[t]$ be relatively prime, $A ∈ M_n(\mathbb{C})$. Show that $\operatorname{rank}(p(A))+\operatorname{rank}(q(A)) ≥ n$.
Let $p(t), q(t) ∈ \mathbb C[t]$ be relatively prime, $A ∈ M_n(\mathbb{C})$. Show that $\operatorname{rank}(p(A))+\operatorname{rank}(q(A)) ≥ n$. I have been stumped on this question for quite awhile. ...