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Matrices Proof> C=A-B, if Ax=Bx where x is nonzero, show C is singular

  1. Feb 7, 2010 #1
    1. The problem statement, all variables and given/known data
    Let A and B be n x n matrices and let C= A - B.
    Show that if Ax=Bx, and x does not equal zero, then C must be singular.


    2. Relevant equations



    3. The attempt at a solution
    Ax-Bx=0
    x(A-B)=0
    x(C)=0
    So, Cx=0

    Does that mean C is singular?
     
  2. jcsd
  3. Feb 7, 2010 #2

    Dick

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    If Cx=0 and x is not the zero vector, then what would C^(-1)(0) be? C0=0 as well. Would it be x or 0? Sure, it means C is singular.
     
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