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WI08det-notes218 - Math 1b Practical Determinants Expanded...

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Math 1b Practical — Determinants February 11, 2009 — Expanded February 18 Square matrices have determinants , which are scalars. Determinants can be intro- duced in several ways; we choose to give a recursive definition. The determinant of a 1 × 1 matrix is the entry of the matrix. Once we have defined the determinant of ( n 1) × ( n 1) matrices, we define the determinant of an n × n matrix A with entries a ij as det( A ) = a 11 det( A 11 ) a 12 det( A 12 ) + . . . + ( 1) n 1 a 1 n det( A 1 n ) . Here A ij denotes the submatrix of A obtained by deleting row i and column j from A . It can be seen inductively that the terms (monomials) that appear in det( A ) are products a 1 ,j 1 a 2 ,j 2 · · · a n,j n where j 1 , j 2 , . . . , j n are 1 , 2 , . . . , n in some order, each with a coefficient of +1 or 1. Such a sequence j 1 , j 2 , . . . , j n may be called a permutation of { 1 , 2 , . . . , n } , and the coefficient of the term a 1 ,j 1 a 2 ,j 2 · · · a n,j n in the determinant expansion of A is called the sign of the permutation. For example, when n = 4, the term a 13 a 21 a 34 a 42 arises as det a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 a 31 a 32 a 33 a 34 a 41 a 42 a 34 a 44 = . . . + (+1) a 13 det a 21 a 22 a 24 a 31 a 32 a 34 a 41 a 42 a 44 + . . . = . . . + (+1)(+1) a 13 a 21 det a 32 a 34 a 42 a 44 + . . . = . . . + (+1)(+1)( 1) a 13 a 21 a 34 det ( a 42 ) + . . . = . . . + (+1)(+1)( 1)(+1) a 13 a 21 a 34 a 42 + . . . . So the sign of 3 , 1 , 4 , 2 is 1. [I know this may appear confusing at first glance.] See the handout Mathematica Session 5. The following rule for computing the sign of a permutation may be extracted from the method illustrated above. [There are other approaches to understanding signs, and you may use any of them.] Given a permutation j 1 , j 2 , . . . , j n , write a sign +1 under j i when j i is in an odd-numbered position when j i , j i +1 , . . . j n are rearranged in increasing numerical order, and a sign 1 when j i is in an even-numbered position. Then the sign of the permutation is the product of the signs under the j i ’s. For example, when j 1 , j 2 , . . . , j n = 2 , 6 , 4 , 1 , 5 , 3, we get 2 6 4 1 5 3 + + + + , so the sign of the permutation is +. The sign under the 4, for example, is + because 4 is in the third (an odd) position when 4 , 1 , 5 , 3 is reordered as 1 , 3 , 4 , 5.
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