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18-determinant

18-determinant - DETERMINANTS I Section 6.1...

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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,π ( i ) = 1 is called a permutation matrix or the pattern of π . An in- version is a pair k < l such that σ ( k ) < σ ( l ). The sign of a permutation π , denoted by ( - 1) σ ( π ) 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. σ (1 , 2) = 0, σ (2 , 1) = 1. σ (1 , 2 , 3) = σ (3 , 2 , 1) = σ (2 , 3 , 1) = 1. σ (1 , 3 , 2) = σ (3 , 2 , 1) = σ (2 , 1 , 3) = - 1. DETERMINANT The determinant of a n × n matrix A is defined as the sum π ( - 1) σ ( π ) A 1 π (1) A 2 π (2) · · · A ( n ) , where π is a permutation of { 1 , 2 , . . ., n } and σ ( π ) 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 . If you have seen some multi-variable calculus, you know that det( A ) is the area of the parallelogram spanned by the column vectors of A . The two vectors form a basis if and only if det( A ) 6 = 0. 3 × 3 CASE. The determinant of A = a b c d e f g h i is aei + bfg + cdh - ceg - fha - bdi corresponding to the 6 permutations of (1 , 2 , 3). Geometrically, det( A
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