The Evaluation of Determinants by Expansion

In this paper, we consider a class of substitution strategies which correspond to various minor expansion algorithms, and show that the usual expansion by minors, a row at a time, is optimal in this class. Motivation. One of the least understood aspects of symbolic formula manipulation is the problem of substitution, particularly the merits of various strategies. A specific example, where substitution is complicated enough to be interesting and where theoretical results can be obtained, is the problem of evaluating a determinant by substitution into the Laplace expansion. Introduction. One obvious possibility for evaluating determinants is substitution into the Laplace expansion, regarded as a polynomial form of n ! terms in n2 indeterminates. The obvious way to substitute into this form requires (n — 1) multiplications per term, a total of (w — l)n! multiplications. Many of these multiplications are redundant, and no advantage is taken of any collection of like terms. Some sort of parenthesizing and grouping of terms, as in Homer's rule, is clearly indicated. Many such groupings are possible. For example, when n = 4, the 4 ! terms XuX22X33Xit XxxX22X3iXi3 -*l l-*23-*32*44 + XxxX23X3iXi2 XXXX2iX33Xi2 + XxxX2iX32Xi3 Xx2X2xX33Xn + Xx2X2xX3iXi3 + *12-*23*31*44 •*12*23*34*41 + -*12-*24-*33*41 Xx2X2aX3\Xa,3 T •*13*2l'*32-*44 Xx3X2xX3iXi2 -*13*22-*31-*44 + *13-*22-*34-*41 + •*13-*24-*31*42 ^13^24^32^41 + ^14^21-^33-^42 -*14-*21.*32-*43 T XxiX22X3lXi3 *14-*22*33-*41 ■*14-*23*31-*42 + -*14-*23*32-*41 can be written, among other ways, as ÍXXXX22 Xx2X2x)ÍX33Xu -X34*43) (*ll-*23 ■^13JC2l)(^32^44 X3aXí2) + (•*ll-*24 -*14.*2l)v*32-*43 x-HXi2) + (*l2.X23 Xx3X22)(x3xXu .*34-X4l) iXx2X2i XxiX22)iX3xXi3 -*33-*4l) + (■*13-*24 Xxa,X23)ÍX3xXa,2 -*32-*4l) Received June 8, 1973. AMS (MOS) subject classifications (1970). Primary 15A15, 68A15, 68A20.