WI08det-notes218

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

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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 defnition. The determinant o± a 1 × 1 matrix is the entry o± the matrix. Once we have defned the determinant o± ( n 1) × ( n 1) matrices, we defne the determinant o± 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 o± A obtained by deleting row i and column j ±rom A . It can be seen inductively that the terms (monomials) that appear in det( A )a r e 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 o± +1 or 1. Such a sequence j 1 2 n may be called a permutation { 1 , 2 } , and the coefficient o± the term a 1 ,j 1 a 2 ,j 2 a n,j n in the determinant expansion A is called the sign o± the permutation. For example, when n =4,theterm 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 ² + = + (+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 o± 3 , 1 , 4 , 2is 1. [I know this may appear con±using at frst glance.] See the handout Mathematica Session 5. The ±ollowing rule ±or computing the sign o± a permutation may be extracted ±rom the method illustrated above. [There are other approaches to understanding signs, and you may use any o± them.] Given a permutation j 1 2 n , write a sign +1 under j i when j i is in an odd-numbered position when j i i +1 ,...j n are rearranged in increasing numerical order, and a sign 1wh en j i is in an even-numbered position. Then the sign o± the permutation is the product o± the signs under the j i ’s. For example, when j 1 2 n = 2 , 6 , 4 , 1 , 5 , 3, we get ± 264153 +++ + ² , so the sign o± the permutation is +. The sign under the 4, ±or example, is + because 4 is in the third (an odd) position when 4 , 1 , 5 , 3 is reordered as 1 , 3 , 4 , 5. The ±ollowing rules are extremely important. Some explanation o± why they hold will be given later, but ±or the moment we just apply them.
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(o) The determinant of the identity matrix is 1.
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This note was uploaded on 09/25/2010 for the course MATH 1bPRAC taught by Professor Wilson during the Winter '09 term at Caltech.

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

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