1 Determinants Eng

# 1 Determinants Eng - 1 APPLICATIONS OF MATRICES AND...

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2 A (adj A ) = | A | I 3 = (adj A ) A In general we can prove that A (adj A ) = | A | I n = (adj A ) A . Example 1.1 : Find the adjoint of the matrix A = a b c d Solution: The cofactor of a is d , the cofactor of b is c , the cofactor of c is b and the cofactor of d is a . The matrix formed by the cofactors taken in order is the cofactor matrix of A . The cofactor matrix of A is = d c b a . Taking transpose of the cofactor matrix, we get the adjoint of A . The adjoint of A = d b c a Example 1.2 : Find the adjoint of the matrix A = 1 1 2 3 2 3 Solution: The cofactors are given by Cofactor of 1 = A 11 = 3 3 = 3 Cofactor of 1 = A 12 = 1 3 3 = 9 Cofactor of 1 = A 13 = 2 2 1 = 5 Cofactor of 1 = A 21 = 1 1 3 = 4 Cofactor of 2 = A 22 = 1 1 2 3 = 1 Cofactor of 3 = A 23 = 1 2 1 = 3 Cofactor of 2 = A 31 = 1 2 3 = 5
3 Cofactor of 1 = A 32 = 1 1 1 3 = 4 Cofactor of 3 = A 33 = 1 1 1 2 = 1 The Cofactor matrix of A is [ A ij ] = 3 9 5 4 3 5 1 adj A = ( A ij ) T = 4 5 9 4 1 Example 1.3 : If A = 2 4 , verify the result A (adj A ) = (adj A ) A = | A | I 2 Solution: A = 2 4 , | A | = 2 4 = 2 adj A = 4 2 1 1 A (adj A ) = 2 4 4 2 1 1 = 2 0 0 2 = 2

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## This note was uploaded on 09/21/2009 for the course CS 580 taught by Professor Fdfdf during the Spring '09 term at University of Toronto.

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1 Determinants Eng - 1 APPLICATIONS OF MATRICES AND...

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