Lect8 supp

Lect8 supp - MATH1111/2010-11/Supplement 1 Objective : In...

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MATH1111/2010-11/Supplement 1 Objective : In this note, we shall introduce an alternative way to evaluate the determinant of a matrix, then we can see why the cofactor expansion along any column or any row is the same. (This is a result assumed without proof in the textbook.) To start with, we need the concept of permutation which should be known to you. (See Chapter 3, Section 1 of [1].) For example, (1 2 3 4) ±! (2 4 3 1) is a permutation of the four objects 1 ; 2 ; 3 ; 4. It can be realized as 1 ! 2, 2 ! 4, 3 ! 3, 4 ! 1. More precisely, we de±ne a permutation as follows: De±nition . A permutation ± of the set f 1 ; 2 ; ²²² ;n g is an one-to-one mapping y from f 1 ; 2 ; ²²² ;n g onto f 1 ; 2 ; ²²² ;n g , i.e. ± is a bijection. In the above example, ± : f 1 ; 2 ; 3 ; 4 g ! f 1 ; 2 ; 3 ; 4 g is given by ± (1) = 2, ± (2) = 4, ± (3) = 3 and ± (4) = 1. For convenience, we put it in the following form: ± = ± 1 2 3 4 2 4 3 1 ² : Now if ² is another permutation of f 1 ; 2 ; 3 ; 4 g , we can consider the composite ² ³ ± of ² and ± (like the usual composite of functions). We write ²±
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Lect8 supp - MATH1111/2010-11/Supplement 1 Objective : In...

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