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given a matrix $B$

for example

$$ B= \begin{pmatrix} 1 & 1 & 1 \\ 2 & 2 & 5 \\ 4 & 6 & 8 \\ \end{pmatrix} $$ I know how to perform the Gaussian elimination w/wo pivoting but I just wanted to know when is it possible to complete the algorithm of elimnation without pivoting till the end cuz I'm coding a program doing that and I want to add a condition in it to make sure that it works correctly

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    There is no easy characterization for "no need of permutation". There are some easy sufficient conditions, like the various versions of diagonal dominance. – 2017-02-22
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    The GE without pivoting is possible iff all the leading principal submatrices are nonsingular. The simplest method to verify this is to actually try to perform the GE. Of course the factorization could be of poor numerical quality unless all diagonal pivots are "good", e.g., as in case of a diagonally dominant matrices. – 2017-02-23

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