This preview shows pages 1–4. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Chapter 2 MAT188H1F Lec03 Burbulla Chapter 2 Lecture Notes Fall 2007 Chapter 2 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 2 Chapter 2 2.2: Determinants and Inverses Chapter 2 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 2 2.2: Determinants and Inverses Determinants of Elementary Matrices The following four slides will establish some results about determinants and elementary matrices that will allow us to prove a major result: If A and B are n n matrices, then det( AB ) = det A det B . Let E be an elementary matrix. 1. If E is obtained from I by switching two rows of I , then det E = = det I = 1 . 2. If E is obtained from I by multiplying a row of I by c 6 = 0 , then det E = c det I = c . 3. If E is obtained from I by adding a multiple of one row of I to another row of I , then det E = det I = 1 . These are special cases of Properties 4, 2 and 6, respectively, of my lecture notes for Section 2.1 Note that for every elementary matrix E , det E 6 = 0 . Chapter 2 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 2 2.2: Determinants and Inverses Lemma: det( EB ) = det E det B , E an Elementary Matrix There are three cases. 1. If E is obtained from I by interchanging two rows of I , then EB is the same as the matrix B , except that two of its rows have been interchanged. Thus det( EB ) = det B . On the other hand, det E det B = ( 1) det B = det B . 2. If E is obtained from I by multiplying a row of I by c 6 = 0 , then EB is the same as the matrix B , except that one of its rows has been multiplied by c . Thus det( EB ) = c det B . On the other hand, det E det B = c det B . 3. If E is obtained from I by adding a multiple of one row of I to another row of I , then EB is the same as the matrix B , except that a multiple of one row of B has been added to another row of B . Thus det( EB ) = det B . On the other hand, det E det B = (1) det B = det B . Two consequences of this Lemma follow. Chapter 2 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 2 2.2: Determinants and Inverses If A C by a series of k elementary row operations, then det C = det E k det E k 1 det E 2 det E 1 det A , for the elementary matrices E 1 , E 2 , . . . , E k 1 , E k that correspond to the k elementary row operations used. Proof: We have A E 1 A E 2 ( E 1 A ) E k ( E k 1 E 2 E 1 A ) = C ....
View
Full
Document
 Fall '07
 Burbula

Click to edit the document details