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Matrix Multiplication Proof

  1. Jan 22, 2010 #1
    1. The problem statement, all variables and given/known data
    I'm doing a proof in which I need to show:
    given that AX = 0, AVX=0 where V is invertible.

    Also, given that AVY = 0, then AY = 0.


    2. Relevant equations



    3. The attempt at a solution
    I can't remember from the previous course I took how to do this. I know that I can multiply from the left or right by V-1, but seeing as V is in the middle that won't work.

    This is part of a larger proof, if it would make more sense to have the entire question let me know.
     
  2. jcsd
  3. Jan 22, 2010 #2
    I'm assuming A,X,V,Y are matrices, but I'm not sure (EDIT: I see you stated in the title that they are matrices). Also do X and Y need to be vectors or are they general matrices. Do we require some matrices to be square or non-zero? Any other assumptions?

    The information you have given is not sufficient. Consider:
    [tex]A = \left[\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right][/tex]
    [tex]X = \left[\begin{array}{cc} 0 \\ 0 \end{array} \right][/tex]
    [tex]V = \left[\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right][/tex]
    [tex]Y = \left[\begin{array}{cc} 1 \\ 0 \end{array} \right][/tex]
    Then [itex]V^2=I[/itex] so V is invertible. AX = AVX = AVY = 0, but,
    [tex]AY = \left[\begin{array}{cc} 1 \\ 0 \end{array} \right][/tex]
     
  4. Jan 22, 2010 #3
    A slightly more interesting example where all matrices are non-zero:
    [tex]A = \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 &0 \\ 0 &0 & 0 \end{array} \right][/tex]
    [tex]X = \left[\begin{array}{c} 0 \\ 0 \\ 1 \end{array} \right][/tex]
    [tex]V = \left[\begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1&0&0 \end{array} \right][/tex]
    [tex]Y = \left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array} \right][/tex]
    Then [itex]V^3=I[/itex] so V is invertible. AX = AVX = AVY = 0, but,
    [tex]AY = \left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array} \right][/tex]
     
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