1 Determinants Eng

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

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1 1. APPLICATIONS OF MATRICES AND DETERMINANTS 1.1. Introduction : The students are already familiar with the basic definitions, the elementary operations and some basic properties of matrices. The concept of division is not defined for matrices. In its place and to serve similar purposes, the notion of the inverse of a matrix is introduced. In this section, we are going to study about the inverse of a matrix. To define the inverse of a matrix, we need the concept of adjoint of a matrix. 1.2 Adjoint : Let A = [ a ij ] be a square matrix of order n . Let A ij be the cofactor of a ij . Then the n th order matrix [ A ij ] T is called the adjoint of A . It is denoted by adj A . Thus the adj A is nothing but the transpose of the cofactor matrix [ A ij ] of A . Result : If A is a square matrix of order n , then A (adj A ) = | A | I n = (adj A ) A , where I n is the identity matrix of order n . Proof : Let us prove this result for a square matrix A of order 3. Let A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 Then adj A = A 11 A 21 A 31 A 12 A 22 A 32 A 13 A 23 A 33 The ( i , j ) th element of A (adj A ) = a i 1 A j 1 + a i 2 A j 2 + a i 3 A j 3 = = | A | if i = j = 0 if i j A (adj A ) = | A | 0 0 0 | A 0 0 | A | = | A | 1 0 0 0 1 0 0 0 1 = | A | I 3 Similarly we can prove that (adj A ) A = | A | I 3
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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
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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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1 Determinants Eng - 1. APPLICATIONS OF MATRICES AND...

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