$AB=A+B$ implies $AB=BA$

Question

This is problem 14 from section 1.2 in the book Matrix Theory Basic Results and Techniques 2nd edition.

If $A B=A+B$ for matrices $A, B,$ show that $A$ and $B$ commute, i.e., $$ A B=A+B \quad \Rightarrow \quad A B=B A $$

The problem appear to be a "simple" problem that we just need to play around with $AB=A+B$. But after a hour of trying, I couldn't figure it out. Any hint will be appreciated.

Do you know that if $XY=I$ then $YX=I?$ If so, let $X=A-I$ and $Y=B-I.$ — Thomas Andrews, Sep 27 '20 at 20:39

score 1 · Answer 1 · answered Sep 27 '20 at 20:40

Hint: $AB=A+B \implies AB-A-B+I=I \implies (A-I)(B-I)=I$, from here prove that $A-I$ and $B-I$ are inverses of each other (note, you can prove directly from the given equation that they are square matrices). After proving that, write $(B-I)(A-I)=I$ and expand.

score 1 · Answer 2 · answered Sep 27 '20 at 20:41

1

Rearrange to $(A-1)(B-1)=1$. So these matrices are inverse of one another ... so $(B-1)(A-1)=1$ and the result follows.

answered Sep 27 '20 at 20:41

Donald Splutterwit

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$AB=A+B$ implies $AB=BA$

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