Two lesser-known classical elimination techniques are applied in matrix form to the linear system stability problem of determining when the roots of a realnth degree polynomial all have negative real parts. In particular, for calculation of the critical Hurwitz determinant Δn−1a new tabular scheme is obtained which is superior to the Routh array for parametric stability problems. An extension leads to new half-order expressions for the other ΔtAnother reduction of Δ−1in terms of minors of order about ¼n is also given.