Sunday, 22 May 2016

linear algebra - If AB=I then BA=I: is my proof right?





I want to prove that for matrices A,BMn(K) where K{R,C,H} if AB=I then BA=I.



My proof is really short so I'm not sure it's right:



If AB=I then (BA)B=B and therefore BA=I?


Answer




The implication (BA)B=BBA=I is a little quick and not always true...



But observe that
1=det(BA)=det(B)det(A)


thus B is invertible and it follows that
BA=BA(BB1)=B(AB)B1=BB1=I.


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