I want to prove that for matrices A,B∈Mn(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=B⇒BA=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(BB−1)=B(AB)B−1=BB−1=I.
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