Even though I was never taught linear algebra fully, to do this problem I would consider what would make the matrix A invertible and what would it mean if the RRE form wasn't the identity matrix.
This isn't too hard to prove. You can start by asking yourself what a row operation on a matrix translates to in matrix algebra. And what do the matrices corresponding to the row-operations amount to when they row-reduce A to I?
As for the "forward" conjecture, well I can think of something some might find objectionable. If it does not row-reduce A to I, it the RRE form has a row of zeros. That means that the determinant is 0 and hence it is not invertible. I'm sure there's a better way to do this.