This definition is from Hoffmann and Kunze.
Definition. If $A$ and $B$ are $m \times n$ matrices over the field $F$, we say that $B$ is row-equivalent to $A$ if $B$ can be obtained from $A$ by a finite sequence of elementary row operations.
Suppose I have two different matrices and I perform 999 elementary row operations on $A$ on paper and cannot get $B$, so I conclude that they are not row equivalent, but the 1000th step (which I did not do) makes them row equivalent.
My question is what is the meaning of "finite sequence of elementary row operations" in the above definition. Is it that Linear algebra is modern so we must resort to computers.