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Given two matrices $A_{m\times n}$ and $B_{n\times p}$, what is the sufficient and necessary condition for $AB$ to have full rank?

I know $r(AB)=r(B)-\dim N(A) \cap R(B)$, so is it true the above iff $\dim N(A)\cap R(B)=0$? But seems incorrect if $m\lt p$. Please help.

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    Related post: [93989](http://math.stackexchange.com/q/93989/). See Robert Israel's answer in that post.2011-12-26
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    @Srivatsan: I don't see how the answer can be directly translated to my question, unless you show explicitly. Thanks.2011-12-26
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    Seems to me that if $n$ is the smallest of the three numbers then full rank is impossible.2011-12-27

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