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Understanding Matrices Sums and Products

  1. Feb 13, 2016 #1
    1. The problem statement, all variables and given/known data
    Suppose that AB = AC for matrices A, B, and C.
    Is it true that B must equal C? Prove the result or find a counterexample.

    2. Relevant equations
    Properties of matrix multiplication

    3. The attempt at a solution
    AC = A(D + B) = AD + AB = 0 + AB = AB ??? Can someone help me understand in plain english?
     
  2. jcsd
  3. Feb 13, 2016 #2

    Ray Vickson

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    If you write AB = AC as A(B-C) = 0, is it true that B-C = 0? That is, does having AD = 0 imply that D must = 0?
     
    Last edited: Feb 14, 2016
  4. Feb 14, 2016 #3

    HallsofIvy

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    IF A has an inverse the AB= AC gives [itex]A^{-1}AB= A^{-1}AC[/itex] so [itex]IB= IC[/itex] so [itex]B= C[/itex] so you question becomes "does every matrix have an inverse?".

    By the way, your question, as stated, is trivially false even for numbers: 0B= 0C for any B and C, it does NOT follow that B= C. To make you question at all interesting you should add "A non-zero".
     
  5. Feb 14, 2016 #4
    Not helpful
     
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