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Prove that if A is an invertible matrix and AB = BC then B = C.

  1. Oct 15, 2009 #1
    Prove that if A is an invertible matrix and AB = BC then B = C. I thought the way to approach it was to use A^-1 on the equality AB=BC but then I got stuck. I can't get it so that B is on one side and C alone is on the other.

    My Try:
    AB = BC
    A^-1 AB = A^-1 BC
    IB = A^-1BC

    Also why does B = C not contradict the statement that "the cancellation law doesn't hold for matrices"?
  2. jcsd
  3. Oct 15, 2009 #2


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    Science Advisor

    Surely, that's not what you want to prove? It's not true. Take A= C and then, for any B, AB= BC because both are just AB.

    You must have meant "If AB= AC then B= C". Just multiply both sides on the left by A[sup[-1[/sup].

    And it does not contradict the statement "the cancellation law doesn't hold for matrices" because the cancellation law says that "If AB= AC for any non-zero A, then B= C". The theorem you are trying to prove requires that A be invertible. There are many non-zero matrices that are not invertible.
    Last edited by a moderator: Oct 15, 2009
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