Simple Commutative Relation on Matrices

Linear Algebra exam problems and solutions at University of California, Berkeley

Problem 55

Let $A$ and $B$ are $n \times n$ matrices with real entries.
Assume that $A+B$ is invertible. Then show that
\[A(A+B)^{-1}B=B(A+B)^{-1}A.\]

(University of California, Berkeley Qualifying Exam)

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Proof.

Let $P=A+B$. Then $B=P-A$.

Using these, we express the given expression in terms of only $A$ and $P$.
On one hand, we have
\[A(A+B)^{-1}B=AP^{-1}(P-A)=AP^{-1}P-AP^{-1}A=A-AP^{-1}A.\] On the other hand we have
\[B(A+B)^{-1}A=(P-A)P^{-1}A=PP^{-1}A-AP^{-1}A=A-AP^{-1}A.\] Thus these are equal.

This completes the proof.

Comment.

Did I make a mistake? Isn’t it too simple to be a qualifying exam problem?


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