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

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