# section22 - Chapter 2 MAT188H1F Lec03 Burbulla Chapter 2...

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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 ....
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section22 - Chapter 2 MAT188H1F Lec03 Burbulla Chapter 2...

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