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The questions is actually same with the title : Prove that a square matrix with either a left or right inverse is invertible.

I didn't quite get the questions to be honest. A square matrix may have right or left inverse or both. But existence of one of the right or left inverse doesn't imply that matrix has both right and left inverse ?

I mean, is it possible to we have a right inverse but not a left ?

Bernard
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Scott
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  • Yes it is. It isn't commutative – Rushabh Mehta Oct 28 '19 at 15:23
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    Non-square matrices can certainly have one inverse and not the other. It turns out that square matrices can't have one without the other, but it's not immediately obvious, and must be proven. – Arthur Oct 28 '19 at 15:25
  • @Arthur That's true, but OP seems to take it to be obvious, since OP views matrices like any other commutative ring. – Rushabh Mehta Oct 28 '19 at 15:47

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