1.4 The inverse of a matrix - The inverse of a matrix...

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The inverse of a matrix Definition. Let A = [ a ij ] be an n×n matrix. Let M ij be the ( n 1 ) × ( n 1 ) sub matrix of A obtained by deleting the i th row and j th column of A . M ij = [ a 11 a 12 a 1 j …a 1 n a 12 a 22 a 2 j …a 2 n …………………… .. a i 1 a i 2 a ij … a ¿ ………………… .. a n 1 a n 2 a n j …a nn ] ithrow j th column The determinant | M ij | is called the minor of a ij . The cofactor A ij of a ij is defined as A ij =(− 1 ) i + j | M ij | . Example 1. Let A = [ 3 1 2 4 5 6 7 1 2 ] . Then | M 12 | = | 4 6 7 2 | = 8 42 =− 34, | M 23 | = | 3 1 7 1 | = 3 + 7 = 10, | M 31 | = | 1 2 5 6 | =− 6 10 =− 16.
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Hence A 12 = ( 1 ) 1 + 2 ( 34 ) = 34, A 23 = ( 1 ) 2 + 3 10 =− 10, A 31 = ( 1 ) 3 + 1 ( 16 ) =− 16. Theorem 1. (Determinant of n order square matrix). Let A = [ a ij ] be a n×n size matrix. Then for each 1 ≤i≤n a i 1 A i 1 + a i 2 A i 2 + + a ¿ A ¿ and for each 1 ≤ j ≤n a 1 j A 1 j + a 2 j A 2 j + + a nj A nj are same . This number is called the determinant of the matrix A and denoted as | A | . The first expression is called the expansion of | A | along the i-th row,
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