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My reasoning is yes, as you can switch row i with row i in the matrix... But I'm not sure if it's a "legal" elementary operation to switch a row with itself.

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This is a question of convention, but I would certainly consider the identity an elementary matrix, as I think most other mathematicians would. It corresponds to the elementary row operation of "doing nothing", which is about as simple as it gets.

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Thank you.. And is it considered "legal" to switch a row with itself? –  Daniel Aug 13 '12 at 21:04
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@Daniel Row/column swapping is the wrong type to file it under :) –  rschwieb Aug 13 '12 at 21:05
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I don' think the identity matrix can be considered as swapping one row with itself, as it does not change the sign of the determinant. However, it should be perfectly legitimate to consider it as adding zero times one row to another row, or multiplying one row with the numbeer one. –  Per Manne Aug 13 '12 at 21:12
    
@Daniel I'm also not sure what you mean by "legal". It's certainly very basic to do nothing. –  Alex Becker Aug 13 '12 at 21:23
    
@AlexBecker Well, I have a homework problem that involves two rows being swapped, i and j. The answer is completely different if I can assume that i can be equal to j, which is why I was wondering if it was allowed. –  Daniel Aug 13 '12 at 21:26
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Sure, it fits very well as a row/column scaling operation scaling rows by 1, (but not really as a swapping operation.)

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The identity matrix is the multiplicative identity element for matrices, like $1$ is for $\Bbb{N}$, so it's definitely elementary (in a certain sense).

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