I believe I have proven that any Type 1 ERO (swapping two rows of a matrix) can be represented with a sequence of Type 2 (apply a scalar multiple to a row) and Type 3 (add a scaled multiple of one row to another).
In an nxn matrix.,
We can replace the Ra <--> Rb (Type 1), where Ra is row a
With the following sequence:
- Rb = Rb + Ra
- Ra = Ra - Rb
- Ra = (-1)Ra
- Rb = Rb -Ra
Which, as far as I can tell, should always result in the two rows being switched, although I feel like this isn't rigorous, or formally proved.
Is this right? If so, how can I prove more formally?