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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:

  1. Rb = Rb + Ra
  2. Ra = Ra - Rb
  3. Ra = (-1)Ra
  4. 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?

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  • $\begingroup$ This is correct and about as rigorous as I would want it to be. In a first course on linear algebra it would make things even clearer for some students if you also listed the contents of the two rows after each operation. But the question is in a sense a duplicate of this. You get away with less steps actually! $\endgroup$ Feb 14, 2018 at 7:12

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