Class Note for MATH 790 at KU
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Date Created: 02/06/15
Determinants Linear Algebra Notes Satya Mandal October 47 2005 1 Generalities We will work with determinants of matrices over a commutative rings Before we do that7 we will want to talk about a modules over a commutative ring A module over a ring is what vector spaces are over a led De nition 11 Let K be a commutative ring A nonempty set M is said to be module over K if 1 M is an abelian group under addition 2 There is a scalar multiplication K X M a M That means given a E K and 6 M there is an element ad 6 M 5 Scalar multiplication is associative and distributive That means for ab 6 K and my 6 M7 we have 1ab dam7 ay adnLay7 ab ab41 x Let K be a commutative ring and M be module over K Then MTMgtltMgtltgtltM will denote cartesian product of ricopies M Remark 11 Let K be a commutative ring and V K Therefore V is a Kimodule For an integer n 2 17 let lMLmK7 or simply Mm denote the group of all m X n matrices with coef cients in K We also write Mn MMK MmK Let A 127 be an n X 72 matrix Let vi Wham 7am be the 1th row of A Note that v1 02 A vn is an element of V We may write A v17 71 as a row instead of a column if conventmet In other words7 V Mm Therefore7 from whatever you know about determinants we have def V a l is a function with various properties We will formalize these propeties of determinats in the following de nitions De nition 12 Let K be a commutative ring and M be a module over K Let D MT a K be a function 1 D is said to be a multilinear function if D is linear in each coordinate That means D17 7 17au bv7 17 7 aD177 17u7 17TbD17 17v7 177m for all ab 6 K and 231171 6 M 2 D is said to be an alternating function if 1 D 7Dy where m 17 7 6 MT and y is found from m by switch ing mi and mj 2 and D 0 whenever m 17 7 6 MT and m 3 for some 239 a j Lemma 11 Let M be a module over a commutative ring K7 and 2 a 0 and 12 E K Then for a function D MT a K the following are equivalent 1 D is alternating 2 D 7Dy where a x177T 6 MT and y is found from a by switching mi and 7 Proof i 2 Obvious 2 i 1 Supose a 6 MT is such that m a for some i a j Then by D 7D Therefore 2D 0 Since 2 a 07 we have 2 is invertible So7 D 0 and the proof is complete De nition 13 Let K be a commutative ring and MnK be the ring of all n X nmatrices Note that MnK M where M K We say that a function DM aK is a determinant function if D is alternating multilinear and also DIn 1 Remark 12 Obviously7 1 Multilinear corresponds to the fact that determinant are linear with respect to each row and 2 alternating corresponds to the property that when you switch two rows the de terminant changes sign Theorem 11 Let K be a commutative ring Then there is a UNIQUE determinat function D MnK a K We will denote this function by detgtk7 or for a matrix A E MnK we will denote DA by detA Proof Use induction on n 2 Usual Facts Theorem 21 Let X be a eld 1 2 For A7 B E MnX we have detAB detA detB For AB 6 MnX if A is equivalent to B ie A P lBP for some P E GLnlF also see page 94 then det A detA For A E MnX we have detA detAt For square matrices X7Y7 W7 Z 6 MnX and lt55 detA detX detY 7 detZ detW A we have For square matrices X E MTXY7E M5X and a r X 3 matrix Z and X Z A lt 0 Y detA detX detY we have For A E MnX7 denote the adjoint of A by adjA Then AadjA adjAA detAIn bf Permutations and determinants Suppose Sn denote the group of permutations of the set 1727772 For 0 E Sn7 de ne signa 1 if 0 is even7 else de ne 39720 71 Now for A E MnX we have n detA Z siyn0ll am 76 Sn 2391 Proof Verify that the right hand side satis es the de nition 13 of the determinant function 3 Determinant of Linear Operators Let V be a vector space over a eld l and dimV 71 Let L V a V be a linear operator Let A be the matrix of T With respect to a basis 617 7en The de ne detT detA Note that by 2 of theorem 217 it follows that detT is well de ned ie does not depend on the basis
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