Why define three elementary row operations? When one of them can be performed by the others? I'm learning about linear algebra and in the course we've defined three "elementary row operations"
$$(1) \text{ Switching any two rows}$$
$$(2) \text{ Non-Zero scaling of any row}$$
$$(3) \text{ Adding a multiple of one row to a different row}$$
However it seems all these operations can be performed by the two simpler operations:
$$(1) \text{ Non-Zero scaling of any row}$$
$$(2) \text{ Adding any two rows}$$
I mean why even have switching rows as an operation in the first place when it can be performed by the addition and scaling of rows? It just seems redundant, why add an extra operation that can already be performed by the other operations?

Edit: Because someone in the comments asked how the swapping of rows could be performed with just non-zero scaling and adding rows. If you wanted to swap, say row $p$ with row $q$ you would:
$$(1) \text{ Add row } q \text{ to row } p$$
$$(2) \text{ Multiply row } p \text{ by } -1$$
$$(3) \text{ Add row } p \text{ to row } q$$
$$(4) \text{ Multiply row } q \text{ by } -1$$
$$(5) \text{ Add row } q \text{ to row } p$$
$$(6) \text{ Multiply row } p \text{ by } -1$$
 A: This is an interesting question, but you didn't go deep enough.  For instance, why do we even have addition, if we can represent adding in terms of set theory?  The answer, as stated above, is simplicity - at a suitable level of abstraction, things become simpler to denote and to describe, talk about, and work with.  Most of the topics in math are somewhat arbitrary decisions to draw a line in the sand and say "here's a new branch, let's develop useful notation and techniques for it".  
The key word is useful.  We use linear algebra because it has great applications both in pure mathematics and applied, and is worthy of study in and of itself.  However, it can definitely be simplified to a considerable degree, if we define "simplification" as "reducing the number of fundamental operations".  If, instead, we define "simplification" as "decrease the effort to use the mathematics by introducing shorthand for common operations", then having all of these elementary row operations is useful.
