On the adjugate matrix

The adjugate AA of a matrix A is the transpose of the matrix of the co-factors of the elements of A. The computation of the adjugate from its definition involves the computation of n2 determinants of order (n−1)—a prohibitively expensive O(n4) process. On the other hand, the computation from the formula AA = det (A)A−1 breaks down when A is singular and is potentially unstable when A is ill-conditioned with respect to inversion. In this paper we first show that the adjugate can be perfectly conditioned, even when A is ill-conditioned. We then show that if due care is taken the adjugate can be accurately computed from the inverse, even when the latter has been inaccurately computed. In Appendix A we give a formal derivation of an observation of Wilkinson on the accuracy of computed inverses.