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Linear Algebra proof

  • Thread starter gotmilk04
  • Start date
  • #1
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Homework Statement


If A is an mXn matrix, B is an nXm matirx, and n<m, then AB is not invertible.


Homework Equations





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.
 

Answers and Replies

  • #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.
 
  • #3
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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.
 
  • #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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