We can think of elementary row operations or elementary matrices as
operations that doesn't change the solution of a system of linear equations...
That's fine with me.. anyhow my question is
"Are permutation matrices or switching rows operation necessary?"
The reason I am asking this is that it seems we can do without them.
If the equations are like
$$y=3, x+2y=2$$
we can switch or we can simply add second equations to the first,
and subtract the first from the second resulting $$x+3y=5, -y=-3.$$
anyhow I am not quite sure that it would apply to more than $2$ equations...