So I have seen that a system of linear equations represented by $Ax = O$ has only the trivial solution. But why does this imply that the augmented matrix [A O] can be rewritten in the form [I O] (using elementary row operations corresponding to $E_1$, $E_2$, . . . , and $E_k$)?
Additionally if $E_k$. . .$E_3E_2E_1A=I$ why does it follow that $A = E_1^{-1}E_2^{-1}E_3^{-}1. . .E_k^{-1}$ (A can be written as the product of elementary matrices).
This is some relevant text to the first paragraph:
I can see that x is a trivial solution when $b = 0$.