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.
Is the identity matrix an elementary matrix?
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linear-algebra
matrices
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0Even if switching a row with itself wasn't for some reason, you're also allowed to multiply a row by a constant. Multiply any row of the identity matrix by 1 and you still get the identity matrix. – 2017-06-02
3 Answers
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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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0Thank you.. And is it considered "legal" to switch a row with itself? – 2012-08-13
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2@Daniel Row/column swapping is the wrong type to file it under :) – 2012-08-13
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1I 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. – 2012-08-13
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0@Daniel I'm also not sure what you mean by "legal". It's certainly very basic to do nothing. – 2012-08-13
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0@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. – 2012-08-13
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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).