Invertible matrix proof

  • Thread starter cmajor47
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  • #1
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Homework Statement


Prove: If A and B are both nxn matrices, A is invertible and AB=BA, then A-1B=BA-1


Homework Equations


(AB)-1=B-1A-1


The Attempt at a Solution


A-1B=BA-1
A-1BB-1=BA-1B-1
A-1BB-1=B(A-1B-1)
A-1(BB-1)=B(BA)-1
B-1A-1(I)=B-1B(BA)-1
(AB)-1=I(BA)-1
(AB)-1[tex]\neq[/tex]A-1B-1

Therefore A-1B does not equal BA-1

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

Answers and Replies

  • #2
gabbagabbahey
Homework Helper
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First, you aren't told that [itex]B^{-1}[/itex] exists, so any proof that relied on the use of [itex]B^{-1}[/itex] is out of the question.

Second, even if [itex]B^{-1}[/itex] did exist then [itex](AB)^{-1}=(BA)^{-1} \Rightarrow AB=BA[/itex] which was already given in the question and so your attempt to find a contradiction has failed.

Third try starting with one of the conditions you're given: [itex]AB=BA[/itex] you're also told that [itex]A^{-1}[/itex] exists, so why not try multiplying both sides of the equation [itex]AB=BA[/itex] by [itex]A^{-1}[/itex] from either side and see what you get...
 
  • #3
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I took your advice and came up with this:

AB=BA
A-1AB=A-1BA
IB=A-1BA
B=A-1BA
BA-1=A-1BAA-1
BA-1=A-1B(AA-1)
BA-1=A-1BI
BA-1=A-1B

Read from right to left, this proves the statement. Is this right?
 
  • #4
Dick
Science Advisor
Homework Helper
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It sure does.
 

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