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571_2.1 - 2.1/2.2 THE DETERMINANT OF A MATRIX AND ITS...

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2.1/2.2 THE DETERMINANT OF A MATRIX AND ITS PROPERTIES KIAM HEONG KWA pp. 87-89 Let A = ( a ij ) be an n × n matrix. The ( n - 1) × ( n - 1) matrix M ij obtained from A by deleting the row and the column that contain the ( i, j ) entry a ij of A is called the minor of a ij . The determinant det( A ) of A is defined inductively on its size by det( A ) = ( a 11 if n = 1 , n j =1 a 1 j A 1 j if n 2 , where the scalar A ij = ( - 1) i + j det( M ij ) is called the cofactor of a ij . In fact, Laplace’s formula det( A ) = n X j =1 a ij A ij = n X i =1 a ij A ij for any i, j ∈ { 1 , 2 , · · · , n } . This is called the Laplace’s formula for computing the determinant. Notation: det( A ) = det( a ij ) = a 11 a 12 · · · a 1 n a 21 a 22 · · · a 2 n . . . . . . . . . a n 1 a n 2 · · · a nn . MATLAB function. det(A) returns the determinant of A . Example 1. Let A = a 11 a 12 a 21 a 22 . Then the minors of the entries of A are M 11 = ( a 22 ) , M 12 = ( a 21 ) , M 21 = ( a 12 ) , and M 22 = ( a 11 ) , Date : June 23, 2011. 1
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2 KIAM HEONG KWA so that the cofactors are A 11 = ( - 1) 1+1 det( M 11 ) = a 22 , A 12 = ( - 1) 1+2 det( M 12 ) = - a 21 , A 21 = ( - 1) 2+1 det( M 21 ) = - a 12 , A 22 = ( - 1) 2+2 det( M 22 ) = a 11 . Hence det( A ) = a 11 a 12 a 21 a 22 = a 11 A 11 + a 22 A 22 = a 11 a 22 - a 12 a 22 . Example 2 (Exercise 2.1.3(c) in the text) . Let A = 3 1 2 2 4 5 2 4 5 . Then the minors of the entries of A are M 11 = 4 5 4 5 , M 12 = 2 5 2 5 , M 13 = 2 4 2 4 , M 21 = 1 2 4 5 , M 22 = 3 2 2 5 , M 23 = 3 1 2 4 , M 31 = 1 2 4 5 , M 32 = 3 2 2 5 , M 33 = 3 1 2 4 , so that the cofactors are A 11 = ( - 1) 1+1 det( M 11 ) = 0 , A 12 = ( - 1) 1+2 det( M 12 ) = 0 , A 13 = ( - 1) 1+3 det( M 13 ) = 0 , A 21 = ( - 1) 2+1 det( M 21 ) = 3 , A 22 = ( - 1) 2+2 det( M 22 ) = 11 , A 23 = ( - 1) 2+3 det( M 23 ) = - 10 , A 31 = ( - 1) 3+1 det( M 31 ) = - 3 , A 32 = ( - 1) 3+2 det( M 32 ) = - 11 , A 33 = ( - 1) 3+3 det( M 33 ) = 10 . Hence det( A ) = a 11 A 11 + a 12 A 12 + a 13 A 13 = 3(0) + 1(0) + 2(0) = 0 . Alternatively, det( A ) = a 21 A 21 + a 22 A 22 + a 23 A 23 = 2(3) + 4(11) + 5( - 10) = 0 . Say more? pp. 89-90 For any n × n matrix A , Theorem 2.1.2 det( A ) = det( A T ) .
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2.1/2.2 THE DETERMINANT OF A MATRIX AND ITS PROPERTIES 3 Proof. The conclusion holds trivial for 1 × 1 matrices. Suppose, in- ductively, that the conclusion holds for all ( n - 1) × ( n - 1) matrices. Then det( A ) = n X j =1 ( - 1) 1+ j a 1 j det( M 1 j ) = n X j =1 ( - 1) 1+ j a 1 j det( M T 1 j ) = det( A T ) , where M ij are the minors of the entries of A . The fact that det( M 1 j ) = det( M T 1 j ) for all j ∈ { 1 , 2 , · · · , n } is a consequence of the inductive supposition. Example 3 (Exercise 2.1.8 in the text) . The main diagonal of an n × n matrix A = ( a ij ) is the totality of the the entries a ii , i = 1 , 2 , · · · , n . Such a matrix is called upper triangular if all entries below the main diagonal are zero, i.e., a ij = 0 for all i > j . On the other hand, such a matrix is called lower triangular if all entries above the main diagonal are zero, i.e., a ij = 0 for all i < j . A matrix is called triangular if it is either upper triangular or lower triangular. Note that A is upper triangular if and only if A T is lower triangu- lar. Due to this observation, it is readily seen that if A is triangular, Theorem 2.1.3 then det( A ) = a 11 a 22 · · · a nn .
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