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Math Help - Help with basic matrix proof

  1. #1
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    Help with basic matrix proof

    "Prove or give a counterexample: If A, B, C are 2x2 matrices, AB = BC, and B is invertible, then A = C."

    I'm greatly lost on how to solve this problem; I've been trying for a very long time. I've played around with numbers enough to convince myself that this is indeed a proof. However, I cannot figure out how to prove it unless A,C are either the identity matrix, the zero matrix, or the inverse of B (or if B is the identity matrix). Any help is appreciated.

    EDIT:
    Turns out there are counterexamples. Solutions were distributed after assignment was turned in :\ .
    Last edited by Fenex; September 18th 2009 at 11:09 AM.
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  2. #2
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    Quote Originally Posted by Fenex View Post
    "Prove or give a counterexample: If A, B, C are 2x2 matrices, AB = BC, and B is invertible, then A = C."

    I'm greatly lost on how to solve this problem; I've been trying for a very long time. I've played around with numbers enough to convince myself that this is indeed a proof. However, I cannot figure out how to prove it unless A,C are either the identity matrix, the zero matrix, or the inverse of B (or if B is the identity matrix). Any help is appreciated.
    If B is invertible then simply multiply both sides of the equation by the inverse of B.

     AB = BC

     ABB^{-1} = BB^{-1}C

     AI = IC
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  3. #3
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    Quote Originally Posted by Mush View Post
    If B is invertible then simply multiply both sides of the equation by the inverse of B.

     AB = BC

     ABB^{-1} = BB^{-1}C

     AI = IC
    How are you able to switch the order of C and B^-1 in the second line? Since matrix multiplication isn't commutative.
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