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Linear Algebra proof

  1. Apr 12, 2009 #1
    1. The problem statement, all variables and given/known data
    If A is an mXn matrix, B is an nXm matirx, and n<m, then AB is not invertible.


    2. Relevant equations



    3. The attempt at a solution
    By doing A is a 2X1 and B is a 1X2, I find that AB is not linearly independent, so it cannot be invertible, but I'm not sure how to show that for all matrices of this nature.
     
  2. jcsd
  3. Apr 12, 2009 #2

    Dick

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    Can you show there is a nonzero vector x such that Bx=0? That would make big problems for AB being invertible. And don't PM people about problems, ok? Just post it on the forums and wait a bit.
     
  4. Apr 12, 2009 #3
    Since n<m, there will be a free variable in the nXm matrix B when reduced to echelon form, correct? So then there is obviously more than the trivial solution.
    I'm still confused as to why that creates a problem for AB being invertible.
     
  5. Apr 12, 2009 #4

    Dick

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    If AB has an inverse (AB)^(-1), then for every x, (AB)^(-1)*ABx=x. What happens if ABx=0 and x is not zero?
     
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