# Linear Algebra: Using Gaussian Elimination to obtain Row Echelon Form Matrix

I am learning Linear Algebra and I understand that we can use Gaussian Elimination to transform an augmented matrix into its Row Echelon Form using Elementary Row Operations. (ERO)

One thing that is not very clear to me is this: When using EROs, are we restricted to only using the rows in the current iteration of the augmented matrix (that is in the process of being transformed?)

For example, consider I have a matrix with 3 rows, Row A, Row B, and Row C. Now, I perform an ERO and my new augmented matrix (Still not in Row Echelon Form) is of the form Row A, Row A+B, Row C.

To further perform EROs on this new augmented matrix, am I limited to using Row A, Row A+B, Row C, or can I still use Row B (Which is no longer in my current matrix but was in a previous iteration)? And is there a reason why we can/cannot reuse it?

You cannot use row $$B$$. If you do, you could do $$A\to A+B$$. Now you have two equal rows, which in the next step gives you a zero row. And you have put your system in a state which is not equivalent to the original one.