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Does a simple Gaussian elimination works on all matrices? Or is there cases where it doesn't work?

My guess is yes, it works on all kinds of matrices, but somehow I remember my teacher points out that it doesn't works on all matrices. But I'm not sure, because I have been given alot of methods, and maybe I have mixed it all up.

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    I'm not sure what counts as Gaussian elimination proper to you, but: if you don't allow swapping (a.k.a. *pivoting*), then yes, there are matrices on which it fails. If you do, then as long as the matrix isn't singular...2011-09-30
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    Maybe he meant that Gaussian elimination fails over arbitrary commutative rings (like the integers) instead of a field.2011-09-30

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