A note on the stability of Gauss-Jordan elimination for diagonally dominant matrices
Peters and Wilkinson [2] state that "it is well known that Gauss-Jordan is stable" for a diagonally dominant matrix, but a proof does not seem to have been published [1]. The present note fills this gap. Gauss-Jordan elimination is backward stable for matrices diagonally dominant by rows and not for those diagonally dominant by columns. In either case it is forward stable.
gauss-jordan algorithm backward error analysis