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Lecture_6 - Lecture Note 6 Dr Jeff Chak-Fu WONG Department...

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Lecture Note 6 Dr. Jeff Chak-Fu WONG Department of Mathematics Chinese University of Hong Kong [email protected] MAT 2310 Linear Algebra and Its Applications Fall, 2007 Produced by Jeff Chak-Fu WONG 1
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D ETERMINANTS 1. Definition and Properties 2. Cofactor Expansion and Applications D ETERMINANTS 2
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Our aim is: 1. to know the notion of a determinant, 2. to study some of its properties, and 3. to calculate the determinant of the given matrix via the reduction to triangular form. D ETERMINANTS 3
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D EFINITION AND P ROPERTIES 1. Permutation 2. Definition of Determinant 3. Properties of Determinants D EFINITION AND P ROPERTIES 4
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Definition: Let S = { 1 , 2 , . . . , n } be the set of integers from 1 to n , arranged in ascending order. A rearrangement j 1 j 2 . . . j n of the elements of S is called a permutation of S . For example, let S = { 1 , 2 , 3 , 4 } . Then 4123 is a permutation of S . It corresponds to the function f : S S defined by f (1) = 4 f (2) = 1 f (3) = 3 f (4) = 2 . D EFINITION AND P ROPERTIES 5
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We can put any one of the n elements of S in the first position, any one of the remaining n - 1 elements in the second position any one of the remaining n - 2 element in the third position, and so on, until the n th position can be filled by last remaining element. Thus, there are n ( n - 1) ( n - 2) · · · 2 · 1 (1) permutations of S ; we denote the set of all permutations of S by S n . The expression in Equation (1) is denoted n ! , n factoral D EFINITION AND P ROPERTIES 6
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We have 1! = 1 = 1 2! = 2 · 1 = 2 3! = 3 · 2 · 1 = 6 4! = 4 · 3 · 2 · 1 = 24 5! = 5 · 4 · 3 · 2 · 1 = 120 6! = 6 · 5 · 4 · 3 · 2 · 1 = 720 7! = 7 · 6 · 5 · 4 · 3 · 2 · 1 = 5040 8! = 8 · 7 · 6 · 5 · 4 · 3 · 2 · 1 = 40 , 320 9! = 9 · 8 · 7 · 6 · 5 · 4 · 3 · 2 · 1 = 362 , 880 . D EFINITION AND P ROPERTIES 7
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Example 1 S 1 consists of only 1! = 1 permutation of the set { 1 } , namely, 1; S 2 consists of 2! = 2 · 1 = 2 permutations of the set { 1 , 2 } , namely, 12 and 21 ; S 3 consists of 3! = 3 · 2 · 1 = 6 permutations of the set { 1 , 2 , 3 } , namely 123 , 231 , 312 , 132 , 213 and 321 . A permutation j 1 j 2 · · · j n of S = { 1 , 2 , · · · , n } is said to an inversion if a large integer j r precedes a smaller j s . A permutation is called even or odd according to whether the total number of inversions in it is even or odd. Thus, the permutation 4132 of S = { 1, 2, 3, 4} has four inversions: 4 before 1, 4 before 3, 4 before 2, and 3 before 2. It is then an even permutation. Remark: 0 inversion is even. D EFINITION AND P ROPERTIES 8
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Example 2 In S 2 , the permutation 12 is even, since it has no inversions; the permutation 21 is odd, since it has one inversion. Example 3 The even permutation in S 3 are 123 (no inversions); 231 (two inversions: 21 and 31); and 312 (two inversions: 31 and 32). The odd permutations in S 3 are 132(one inversion: 32); 213 (one inversion: 21); and 321 (three inversions: 32,31,21).
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