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For the inverse of a matrix, we learned that to prove two matrices are inverses of each other, you must show that $AB = I = BA$. However, today in class someone claimed you only have to show either $AB = I$ or $BA = I$ to prove they are inverse. I know the commutative property doesn't apply to matrices, so can someone provide an example where $AB = I \ne BA$?

Edit: The question type goes as follows: Given matrix $A$ and matrix $B$, show whether or not the two are inverses. -For this question, do we need to show both $AB = I$ and $BA = I$ or will one or the other suffice?

user378746
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  • The commutative property does not apply to matrices in general however a square matrix will always commute with its inverse. Therefore if you show that either $AB=I$ or $BA=I$ then the other is true by the commutative property. –  Oct 15 '16 at 01:17
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    So you're asking for a counterexample to the claim made by your classmate. But, the classmate's statement is correct for square matrices, so there is no counterexample (for square matrices). – littleO Oct 15 '16 at 01:19
  • Can we consider infinite order matrices? – Oscar Lanzi Oct 15 '16 at 01:43

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Taken from User71352: "Since $AB=I$ then $B=B(AB)=(BA)B.$" Since a matrix times its identity results in the same matrix, $B = BI = B(AB)$ given that $AB = I$. Since matrices follow the Associative Property, $B(AB) = (BA)B$. This tells that $B=(BA)B$. Since only the identity times the matrix will result in the same matrix, $BA$ must equal $I$. Thus, you only need to show $AB = I$ to show the two matrices are inverses.

user378746
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  • Where does $(BA)BB$ come from? How does the conclusion follow from $B=(BA)B$? What are the intermediate steps for? How are you using the fact that the matrices are square? – littleO Oct 15 '16 at 01:38
  • I edited my answer! Thank you! These questions helped me myself to think and understand it. – user378746 Oct 15 '16 at 04:19