The Infinite Gauss-Jordan Elimination on Row-Finite omega x omega Matrices

The Gauss-Jordan elimination algorithm is extended to reduce a row-finite $omegatimesomega$ matrix to lower row-reduced form, founded on a strategy of rightmost pivot elements. Such reduced matrix form preserves row equivalence, unlike the dominant (upper) row-reduced form. This algorithm provides a constructive alternative to an earlier existence and uniqueness result for Quasi-Hermite forms based on the axiom of countable choice. As a consequence, the general solution of an infinite system of linear equations with a row-finite coefficient $omegatimesomega$ matrix is fully constructible.