Gauss Jordan elimination reduces to row-echelon form always? I am reading this text:

and I'm wondering if gauss-jordan elimination always leads to an identity matrix on the left? If so, that helps me understand this passage:

I'm trying to figure out why [A 0] can be rewritten as [I 0]. Why is this?

Gauss-Jordan eliminition works if and only if an inverse exists. It doesn't work for the null element matrix (matrix of zeros).
They said that [A 0] can be rewritten as [I 0] using elementary row operations so there are operations so that works.
