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 1 (Main properties of n × n determinants) Let because the factor in the numerator in the right hand side is precisely det(A) = ? Slide 10
reasoning that shows that the product of determinant factors comes out the same no matter This theorem is imporant for all sorts of reasons
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-
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
Theorem 4 7 A square matrix A is invertible if and only if det(A) is nonzero This last theorem is one that we use repeatedly in the remainder of this text
determinants stated in Theorem 3 1 2 but for rows only (see Lemma 3 6 2) This is clear by property 1 because the row of zeros has a common factor u = 0