The following lemma ensures that the elementary Gaussian operations indeed do not change the solutions of an SLE:

Lemma: Elementary Gaussian Operations Do Not Change the Solutions of an SLE

Let an SLE be given in the form of an extended coefficient matrix $A|\gamma$ and let $S$ be the set of its solutions. Then $S$ is an invariant of the elementary Gaussian operations. In other words, if we apply any finite series of these operations $o_1,o_2,\ldots,o_k$ to $A|\gamma$, leading to the new SLEs

then the solutions $S$ of SLE$_k$ (if any), will be exactly the same as the solutions as the original SLE.

Proofs: 1

Definitions: 1

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  1. Knabner, P; Barth, W.: "Lineare Algebra - Grundlagen und Anwendungen", Springer Spektrum, 2013