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## Homework Statement

Prove: If A and B are both nxn matrices, A is invertible and AB=BA, then A

^{-1}B=BA

^{-1}

## Homework Equations

(AB)

^{-1}=B

^{-1}A

^{-1}

## The Attempt at a Solution

A

^{-1}B=BA

^{-1}

A

^{-1}BB

^{-1}=BA

^{-1}B

^{-1}

A

^{-1}BB

^{-1}=B(A

^{-1}B

^{-1})

A

^{-1}(BB

^{-1})=B(BA)

^{-1}

B

^{-1}A

^{-1}(I)=B

^{-1}B(BA)

^{-1}

(AB)

^{-1}=I(BA)

^{-1}

(AB)

^{-1}[tex]\neq[/tex]A

^{-1}B

^{-1}

Therefore A

^{-1}B does not equal BA

^{-1}

I've been struggling with proofs and was wondering if this was correct?