mat 2310 0607_2_note2

mat 2310 0607_2_note2 - Lecture Note 2 Jan 22 24 2007 Dr...

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Lecture Note 2: Jan 22 - 24, 2007 Dr. Jeff Chak-Fu WONG Department of Mathematics Chinese University of Hong Kong [email protected] MAT 2310C Linear Algebra and Its Applications Spring, 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 . D EFINITION AND P ROPERTIES 5
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D EFINITION OF D ETERMINANT D EFINITION OF D ETERMINANT 6
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Definition: Let A = [ a ij ] be an n × n matrix. We define the determinant of A (written det( A ) or | A | ) by det( A ) = | A | = X ( ± ) a 1 j 1 a 2 j 2 . . . a nj n , (1) where the summation ranges over all permutations j 1 j 2 . . . j n of the set S = { 1 , 2 , . . . , n } . D EFINITION OF D ETERMINANT 7
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P ROPERTIES OF D ETERMINANTS P ROPERTIES OF D ETERMINANTS 8
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Theorem 0.1 The determinants of a matrix and its transpose are equal, that is det( A T ) = det( A ) . P ROPERTIES OF D ETERMINANTS 9
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Theorem 0.2 If matrix B results from matrix A by interchanging two rows (columns) of A , then det( B ) = - det( A ) . P ROPERTIES OF D ETERMINANTS 10
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Theorem 0.3 If two rows (columns) of A are equal, then det( A ) = 0 . P ROPERTIES OF D ETERMINANTS 11
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Theorem 0.4 If a row (column) of A consist entirely of zeros, then det( A ) = 0 .
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