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Invertible matrices problem

  1. Feb 28, 2010 #1
    1. The problem statement, all variables and given/known data
    For the invertible matrices A, B and A-B, simplify the expression [tex](A - B)^{-1}A(A^{-1} - B^{-1})B[/tex].


    2. Relevant equations
    properties of invertible matrices


    3. The attempt at a solution
    [tex](A - B)^{-1}A(A^{-1} - B^{-1})B[/tex]
    = [tex](A - B)^{-1}(AA^{-1}B - AB^{-1}B)[/tex]
    = [tex](A - B)^{-1}(IB - AI)[/tex]
    = [tex](A - B)^{-1}(B - A)[/tex]
    = [tex]I[/tex]

    Am I correct?
     
  2. jcsd
  3. Feb 28, 2010 #2

    vela

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    Almost. The last step isn't quite correct.
     
  4. Feb 28, 2010 #3

    LCKurtz

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    Verrrry close. Check your last step.
     
  5. Feb 28, 2010 #4
    So it just ends here?

    [tex](A - B)^{-1}(B - A)[/tex]
     
  6. Feb 28, 2010 #5

    vela

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    No, you can simplify it. Consider this. Let C=B-A. Then C-1=(B-A)-1, but in your expression, you have (A-B)-1, which isn't the same matrix.
     
  7. Feb 28, 2010 #6
    Ok, so then the answer is:

    [tex]-I[/tex] ?
     
  8. Feb 28, 2010 #7

    LCKurtz

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  9. Feb 28, 2010 #8
    Thanks
     
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