The three elementary row operations are:
- Row switching
- Row multiplication
- Row addition
Each of these actions is a, in a sense, a 'do nothing' action on a linear system of equations. They can be applied without affecting the solution to the equation. It follows too, that if each of these actions 'does nothing', repeated application of them also doesn't affect the solution, so all compositions of elementary row operations also 'do nothing'. My question is, can all possible 'do nothing' actions be written as a composition of these elementary row operations?
Furthermore, my understanding is that row switching can be expressed using only Row multiplication (2) and Row addition (3). What would be the smallest list of 'atoms' (i.e. matrices which can't be composed from simpler matricies) to create all do nothing actions? Is Row addition sufficient to create all 'do nothing' matrices, and how could one show that no other 'do nothing' matrices exist?