A question about an infinite sequence of elementary row operations Do there exist matrices $A$ and $B$ such that $B$ can be transformed into $A$ only if an infinite number of elementary row operations are performed on $B$?
"What can we multiply the top equation by so that we can add it to the bottom equation and eliminate the variable?"
This was the substance of my first three lectures in my college linear algebra class. I was bored, so I came up with this question.
Most importantly:Is this a 'thing' that is already out there(probably is)? What is (or could be) the significance of the sequence of row operations in this context?
Thank you for your time and effort. I look forward to your answers.

Let $A$ be a matrix consisting of all zeros and $B$ be the identity matrix. Then by multiplying each row in $B$ by $1/n$ for any $n>1$ infinitely many times, we can transform it into $A$. However, it is clearly impossible to transform $B$ into $A$ in finitely many steps.
