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So I think I have this proof. Please let me know if it works:

1) The column vectors of a matrix Q are linearly independent

2) rref(Q) is the identity matrix

3) A matrix is invertible if and only if the rref of the matrix is the identity matrix

4) Therefore, Q is invertible

Please let me know if there are other, more elegant proofs.

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That's good enough. – Mlazhinka Shung Gronzalez LeWy Nov 12 '13 at 2:18
    
This is, of course, for a square matrix. – Robert Israel Nov 12 '13 at 2:20
    
Really! :) I thought for sure premise 3 would get me in trouble. – Cormano Nov 12 '13 at 2:34

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