  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 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 $\omega\times\omega$ matrix is fully constructible.
