Linear Algebra: Proving AB Not Invertible for mXn Matrix

[gotmilk04]

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.

 

[Dick]

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.

 

[gotmilk04]

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.

 

[Dick]

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?