In my Hoffman & Kunze book on Linear Algebra, we define 3 elementary row operations, the first of which is the swapping of two rows.
I'm wondering why we need to even have such an elementary operation since it can be achieved through elementary operations of the second and third kind. Is it simply for convenience & simplicity, or is there some deeper reason? It seems that all the linear algebra books that i have checked always include all 3 operations, though one of them is redundant.
I was under the impression that when we try and list the core properties for an object in mathematics the goal is to always minimize the number of properties per definition.