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My linear algebra text defines the elementary row operations on matrices to be:

  1. Swapping rows
  2. Scaling a row by a nonzero scalar
  3. Adding a multiple of one row to a different row

I was thinking it would be simpler to change 3 to just "adding one row to another." It seems unnecessary to have this operation also mention scaling because you could always scale a row by k, add it to some other row, then scale again by 1/k to restore the original values.

Then I remembered that the text also defines elementary matrices as being those that can be produced by doing only one elementary row operation on the identity matrix. My definition is more strict from this perspective -- in effect letting you flip a single 0 to a 1 but not setting it to arbitrary numbers like the original would.

Is this the motivation for defining elementary row operations the way they are? Is there some other reason? For the level of my text I don't really think it's actually that important -- we care about knowing that two matrices being row equivalent means there is a series of elementary matrices that transforms one to the other, but that should still be true with my definition, you just need more matrices to apply and undo the scaling.

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    $\begingroup$ Maybe for convenience, while still keeping the individual operations simple. (Note that swapping rows is also redundant: you can swap rows $i$ and $j$ by adding row $i$ to row $j$, then subtracting row $j$ from row $i$, then adding row $i$ to row $j$, then multiplying row $i$ by $-1$.) $\endgroup$ – Daniel Schepler Sep 13 '17 at 22:57
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    $\begingroup$ Also, for example if you wanted to calculate a determinant using Gaussian elimination, each instance of #3 doesn't change the determinant, while #1 and #2 do, so you might want to minimize the number of times you perform #1 or #2. $\endgroup$ – Daniel Schepler Sep 13 '17 at 22:59

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