What makes elementary row operations "special"?
They're not special. They're just convenient. It's relatively easy to tell what happens to a matrix when you apply an elementary row operation to it, and this isn't quite as true for more complicated types of operations.
In the language of group theory, elementary matrices form a set of generators for the group of invertible square matrices. You could choose a different set of generators if you wanted to, but again, the elementary matrices are convenient.
Performing them preserves the solution space of any associated linear system. They reflect three ways in which we can manipulate a linear system and not change its solution
- Permute two equations.
- Take one equation, add a multiple of another to it and replace it with that.
- Multiply any row by a nonzero scalar.
By the way, it may be amusing to note that you don't really need the row-interchange operation. Thus, instead of having a special operation to interchange rows $i$ and $j$, you could:
- add row $j$ to row $i$
- subtract row $i$ from row $j$
- multiply row $j$ by $-1$
- subtract row $j$ from row $i$
and it would have the same effect. In terms of the elementary matrices,
$$ \pmatrix{1 & -1\cr 0 & 1\cr} \pmatrix{1 & 0\cr 0 & -1\cr} \pmatrix{1 & 0\cr -1 & 1\cr} \pmatrix{1 & 1\cr 0 & 1\cr} = \pmatrix{0 & 1\cr 1 & 0\cr}$$
Similarly, you don't need to be able to add an arbitrary nonzero multiple of row $j$ to row $i$, as long as you can add row $j$ to row $i$: to add $t$ times row $j$ to row $i$, you first multiply row $j$ by $t$, then add row $j$ to row $i$, then multiply row $j$ by $1/t$.
So you could get by with just
- add a row to another row
- multiply a row by a nonzero scalar
Not that I'd want to do this in practice...