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

Question

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?

Many thanks in advance!

Answer 1

In your example, you can continue from Row A, Row A+B, Row C. In this particular example, using Row B instead of Row A+B is a step backwards. There is no reason to do that.

I don't know how far you have studied into this subject, but there is an elementary matrix associated with each Elementary Row Operation. At each step of your operation, the interim matrix that you calculate is the result of multiplication of that step's elementary matrix by the previous step's (interim) matrix. In this way, as you perform a series of Elementary Row Operations, you are in fact multiplying a series of elementary matrices into your original matrix. I hope this gives you a sense why you can't cut back into the middle of the chain of operations and continue from one of the previous steps.

Answer 2

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.