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$\def\rank{\operatorname{rank}}$ Suppose that matrix $A \in \mathbb{R}^{m \times n}, m<n$ is a full row rank matrix, while $B \in \mathbb{R}^{n \times m}$ is a full column rank matrix. Is product $AB$ a regular matrix?

I have experienced this problem while trying to solve a linear system $b = ABx + c$, where $x$ is unknown $m$-dimensional vector, while $A$ and $B$ are matrices with their properties stated above.

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  • $\begingroup$ I tried to give context of my problem and I added a tag, to try to remove "off-topic" problem. $\endgroup$
    – Tarik
    Oct 31, 2014 at 14:01

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Take $A=(1,1)$ and $B=\left(\begin{array}{r}1 \\ -1\end{array}\right).$ Both matrix have rank $1.$ Is it regular the matrix $AB=(0)?$

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  • $\begingroup$ Your example shows it is not valid in general. However, now I do not understand what is wrong with the answer from Sami Ben. $\endgroup$
    – Tarik
    Oct 31, 2014 at 13:15
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If $\exists v\in null(A):\exists x:Bx=v$, then $rank(AB)<m$, else $rank(AB)=m$.

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  • $\begingroup$ Minor point: You should exclude $v=0$ from $null(A)$. $\endgroup$
    – hardmath
    Oct 31, 2014 at 14:45

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