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:
$$(1) \text{ Add row } q \text{ to row } p$$ $$(2) \text{ Multiply row } p \text{ by } -1$$ $$(3) \text{ Add row } p \text{ to row } q$$ $$(4) \text{ Multiply row } q \text{ by } -1$$ $$(5) \text{ Add row } q \text{ to row } p$$ $$(6) \text{ Multiply row } p \text{ by } -1$$