The property states, "A square matrix A is invertible iff it can be written as the product of elementary matrices"
I'm confused on the part of the theorem where they're trying to show that if A is invertible, then it can be written as the product of elementary matrices.
This is that section of the proof:
"Assume A is invertible. You know the system of linear equations represented by Ax=0 has only the trivial solution. But this implies that the augmented matrix [A 0] can be rewritten in the form [I 0] (using elementary row operations corresponding to E1,E2,...,Ek). So, Ek,...,E2,E1A I and it follows that A = E1-1E2-1...Ek-1 . A can be written as the product of elementary matrices."
I just don't get how knowing that Ax=0 has only the trivial solution implies that [A 0] can be written in the form [I 0]. Wasn't it already obvious that A can be rewritten as I since it's invertible? And obviously if there's a 0 matrix adjoined A to it it's going to stay the zero matrix no matter what row operations are done on it? What's the point of doing that?
I'm just generally confused on this proof