18-determinant

18-determinant - DETERMINANTS I Math 21b, O. Knill Section...

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Unformatted text preview: DETERMINANTS I Math 21b, O. Knill Section 6.1: 8,18,34,40,44,42*,56* PERMUTATIONS. A permutation of { 1 , 2 , . . . , n } is a rearrangement of { 1 , 2 , . . . , n } . There are n ! = n ( n- 1) ... 1 different permutations of { 1 , 2 , . . . , n } : fixing the position of first element leaves ( n- 1)! possibilities to permute the rest. EXAMPLE. There are 6 permutations of { 1 , 2 , 3 } : (1 , 2 , 3) , (1 , 3 , 2) , (2 , 1 , 3) , (2 , 3 , 1) , (3 , 1 , 2) , (3 , 2 , 1). PATTERNS AND SIGN. The matrix A with zeros everywhere except A i, ( i ) = 1 is called a permutation matrix or the pattern of . An in- version is a pair k < l such that ( k ) < ( l ). The sign of a permutation , denoted by (- 1) ( ) is (- 1) for an odd number of inversions in the pattern, otherwise, the sign is 1. (To get the sign in the permutations to the right, count the number of pairs of black squares, where the upper square is to the right). EXAMPLES. (1 , 2) = 0, (2 , 1) = 1. (1 , 2 , 3) = (3 , 2 , 1) = (2 , 3 , 1) = 1. (1 , 3 , 2) = (3 , 2 , 1) = (2 , 1 , 3) =- 1. DETERMINANT The determinant of a n n matrix A is defined as the sum (- 1) ( ) A 1 (1) A 2 (2) A n ( n ) , where is a permutation of { 1 , 2 , . . . , n } and ( ) is its sign. 2 2 CASE. The determinant of A = a b c d is ad- bc . There are two permutations of (1 , 2). The identity permutation (1 , 2) gives A 11 A 12 , the permutation (2 , 1) gives A 21 A 22 ....
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This note was uploaded on 04/06/2008 for the course MATH 21B taught by Professor Judson during the Spring '03 term at Harvard.

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