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Prove that if $A,B$ are any such matrices such that $AB$ exists, then $\operatorname{rank}AB \leq \operatorname{rank}A,\operatorname{rank}B$.

I came across this exercise while doing problems in my textbook, but am not sure where to start for the proof of this. I think columnspace might be involved in the proof, although I am not sure.

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marked as duplicate by Andrés E. Caicedo, Amzoti, Potato, amWhy, Pedro Tamaroff Jun 14 '13 at 3:14

This question was marked as an exact duplicate of an existing question.

What is your definition of rank? – Andrés E. Caicedo Jun 14 '13 at 3:04
Rank is the amount of leading 1s in Reduced Row Echelon Form of a matrix. – Sujaan Kunalan Jun 14 '13 at 3:06
Oh, I didn't realize "rankA, rankB" was synonymous with "min(rank(A),rank(B)). I thought the question was saying that rank(AB) was less than or equal to both rank A, and rank B. I must've misread. – Sujaan Kunalan Jun 14 '13 at 4:57
up vote 0 down vote accepted

I think it is best to prove two things that are stronger statements:

$x\in \ker B \Rightarrow x\in \ker AB$,

$y\in \text{im}\;AB \Rightarrow y\in \text{im}\;A$.

Together with the rank-nullity theorem, I believe this provides a solution not in the link @Amzoti provided above. Edit- nevermind, it's in there, but not all in one place.

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