Sylvester’s identity and multistep integer-preserving Gaussian elimination

A method is developed which permits integer-preserving elimination in systems of linear equations, A X = B AX = B , such that ( a ) (a) the magnitudes of the coefficients in the transformed matrices are minimized, and ( b ) (b) the computational efficiency is considerably increased in comparison with the corresponding ordinary (single-step) Gaussian elimination. The algorithms presented can also be used for the efficient evaluation of determinants and their leading minors. Explicit algorithms and flow charts are given for the two-step method. The method should also prove superior to the widely used fraction-producing Gaussian elimination when A A is nearly singular.