I am trying to find two matrices $A$ and $B$ such that matrix $AB$ that is invertible but $BA$ is not.

Have you got any ideas of easy examples? Thank you!

  • $\begingroup$ As has been suggested in Hagen's answer, square matrices aren't going to get you there. I suggest you think in terms of linear mappings between two spaces, one larger than the other. It's easy to map from a small space to a larger space and back to the smaller space without losing information, but what about the other way around? $\endgroup$ – G. H. Faust May 25 '15 at 13:32
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    $\begingroup$ For square matrices, use determinants to see that we cannot do it. $\endgroup$ – GEdgar May 25 '15 at 14:27

$A=\begin{bmatrix}a&a'\end{bmatrix}$, $B=\begin{bmatrix}b\\b'\end{bmatrix}$. Choose $a, a',b,b'$ such that $ab+a'b'\neq 0$. Then $AB=\begin{bmatrix}ab+a'b'\end{bmatrix}$ is invertible, but $$\det(BA)=\det\biggl(\begin{bmatrix}ab&a'b\\ab'&a'b'\end{bmatrix}\biggr)=0.$$


Hint: Nobody said that $A,B$ be square matrices.

  • $\begingroup$ I'm sorry, I forgot to include that both $AB$ and $BA$ have to be defined. $\endgroup$ – marco11 May 25 '15 at 13:39
  • $\begingroup$ Hint: if A is m by n, and B is n by m. then both AB and BA are defined. $\endgroup$ – Mark L. Stone May 25 '15 at 14:11

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