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The three elementary row operations are:

  1. Row switching
  2. Row multiplication
  3. 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?

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  • $\begingroup$ You can't get row multiplication by $\pi$, say, as addition of a row to itself. $\endgroup$ – Gerry Myerson Sep 11 '20 at 4:03
  • $\begingroup$ "It seems like you could write row multiplication as addition of a row with itself" Pray, tell, how many times do you add the row $(10,0,0)$ to itself to get $(1,0,0)$? $\endgroup$ – JMoravitz Sep 11 '20 at 4:03
  • $\begingroup$ Any thoughts on the answer I posted yesterday, Caleb? $\endgroup$ – Gerry Myerson Sep 13 '20 at 3:58
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Yes, all "do nothing" operations are compositions of elementary row operations. If two systems are equivalent, then you can use elementary row operations to bring the first system to reduced row-echelon form, and then use more elementary row operations on that reduced row-echelon form to get to the second system, since the inverse of an elementary row operation is also an elementary row operation.

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