In this section, several theorems about determinants are derived p(a1) = 0, so we have p(x)=(x?a1)p1(x) by the factor theorem (see Appendix D)
(ii) Co-factor of an element aij is given by Aij = (–1)i+j Mij (iii) Value of determinant of a matrix A is obtained by the sum of products of elements of a row
Why are combinatorialists so fascinated by determinant evaluations? A simplistic answer to this question goes as follows Clearly, binomial coefficients ( n k )
Theorem 2 (Determinants and elementary row operations) Let A be a n × n matrix • Let B be the result of adding to a row in A a multiple of another row in A
A set V of positive integers is said to be factor closed (FC) iff all positive factors of any element of V belong to V The following structure theorem and
which contains such factors It is now proposed to examine a direct method for finding the algebraic composition of the determinant M of the coeffi-
is by choosing all the x2s in the linear factors x2 ?xj with 2 < j Thus, this monomial has coefficient +1 in the product In the determinant, the only way to
2 1 Determinants by Cofactor Expansion Since a common factor of any row of a matrix can be From Theorem 2 2 4, the determinants of the elementary
Theorem If A is a square matrix containing a row (or column) of zeros, then det(A) = 0 Proof Use summation of n terms, each term being a product of n factors