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I was trying to find non-invertible matrices $A$ and $B$ such that $AB$ is invertible. I should choose $A$ and $B$ not to be square, otherwise $AB$ invertible would imply $A$ and $B$ invertible too.

All examples I could find were where $A$ is $n\times m$ and $B$ is $m\times n$ where $n<m$.

Is it possible to find $A$ with dimension $n\times m$ and $B$ with $m\times n$ such that $n>m$ and $AB=I$ or $AB$ invertible?

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Use $rank (AB)\leq \min\{rank (A),rank(B)\}$ and that when a matrix has full row rank it has right inverse.

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  • $\begingroup$ Is this true: A has left inverse if and only if A has full row rank? $\endgroup$ – Silent Nov 4 '17 at 6:56
  • $\begingroup$ Also I think you mean full row rank implies right inverse, my book says $A$ right invertible if there is $C$ such that $AC=I$ $\endgroup$ – Silent Nov 4 '17 at 7:08
  • $\begingroup$ @Silent yeah it's a typo. $\endgroup$ – Abishanka Saha Nov 4 '17 at 7:16
  • $\begingroup$ Please answer this: is this true: If $A$ has right inverse, then $A$ has full rank. $\endgroup$ – Silent Nov 4 '17 at 7:54

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