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Linear Algebra Proof using Inverses

  1. Feb 18, 2015 #1

    B18

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    1. The problem statement, all variables and given/known data
    Prove that if A, B, and C are square matrices and ABC = I, then B is invertible and B−1 = CA.
    2. Relevant equations


    3. The attempt at a solution
    I think I have this figured out, just checking it. Heres what I got:
    ABC=I
    (ABC)B-1=IB-1
    (B*B-1)AC=IB-1
    I*AC=IB-1 Cancel I using left hand cancellation property
    AC=B-1
    Thus B-1=CA

    Is every thing I've done here mathematically correct?
     
  2. jcsd
  3. Feb 18, 2015 #2

    LCKurtz

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    How do you know ##B## has an inverse to use? You are trying to prove that.

    And, even if you did, how did you get that step? Matrix multiplication isn't commutative.
     
  4. Feb 18, 2015 #3

    B18

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    Ok, yes I see what you're saying. I can't do the steps I did there. I know that B has to have an inverse because A,B, and Care square matrices and their product is the identity matrix.
     
  5. Feb 18, 2015 #4

    B18

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    Is this a correct path to go down on this proof?
    We have ABC=I
    (AB)C=I. Since (AB)C=I we know that (AB) and C are both invertible. Also this tells us that C=(AB)-1, and (AB)=C-1
     
  6. Feb 18, 2015 #5

    RUber

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    You can also reorder the multiplication using
    CABC = CI = C =IC
    Implies CAB = I.
    Same logic as in your last post should bring you to the solution you are looking for.
     
  7. Feb 18, 2015 #6

    B18

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    How does this look:
    We have ABC=I
    C(ABC)=CI
    CABC=C
    (CABC)A=CA
    (CAB)CA=CA This implies that CAB=I
    CA(BCA)=CA This implies that BCA=I
    CAB=BCA
    (CA)B=B(CA) Then B must be invertible
    Therefore BCA=I
    CA=B-1
     
  8. Feb 18, 2015 #7

    RUber

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    Looks good to me. You hit all the important points.
     
  9. Feb 18, 2015 #8

    LCKurtz

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    I think you need to flesh out your argument with a few more details. Your steps may be correct, but if this is a homework problem you need to fill in some reasons.

    Why must B be invertible? That statement by itself doesn't imply it.

    Why the "therefore" now? Didn't you already have BCA=I above?

    Like I said above, your statements may be true, but your teacher is going to want to know if you know why they are true.
     
    Last edited: Feb 18, 2015
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