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Lect8 - Chapter 2 Determinants Math1111 Determinants...

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Chapter 2. Determinants Math1111 Determinants Motivation Motivation : 1 × 1 case : When is A = ( a ) invertible? Ans: a = 0 . 2 × 2 case : Let A = a 11 a 12 a 21 a 22 . A is nonsingular if and only if A is row equivalent to I if and only if a 11 a 22 - a 21 a 12 = 0
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Chapter 2. Determinants Math1111 Determinants Motivation Motivation : 1 × 1 case : When is A = ( a ) invertible? Ans: a = 0 . 2 × 2 case : Let A = a 11 a 12 a 21 a 22 . A is nonsingular if and only if A is row equivalent to I if and only if a 11 a 22 - a 21 a 12 = 0 Determinant is a number associated to a square matrix, whose value tells whether the matrix is invertible.
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Chapter 2. Determinants Math1111 Determinants Motivation Motivation : 1 × 1 case : When is A = ( a ) invertible? Ans: a = 0 . 2 × 2 case : Let A = a 11 a 12 a 21 a 22 . A is nonsingular if and only if A is row equivalent to I if and only if a 11 a 22 - a 21 a 12 = 0 Determinant is a number associated to a square matrix, whose value tells whether the matrix is invertible. Notation : det ( A ) , det A , | A |
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Chapter 2. Determinants Math1111 Determinants Definition (order 1) Determinant is defined inductively. Det of order 1 Det of order 2 Det of order 3 → ···
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Chapter 2. Determinants Math1111 Determinants Definition (order 1) Determinant is defined inductively. Det of order 1 Det of order 2 Det of order 3 → ··· Definition of determinant of order n : 1 n = 1 . i.e A = ( a ) . Define det A = a .
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Chapter 2. Determinants Math1111 Determinants Definition (order 1) Determinant is defined inductively. Det of order 1 Det of order 2 Det of order 3 → ··· Definition of determinant of order n : 1 n = 1 . i.e A = ( a ) . Define det A = a . 2 n 2 . We need to first define minor and cofactor .
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Chapter 2. Determinants Math1111 Determinants Minor Definition (Minor) Let A = a 11 ··· . . . a 1 j . . . ··· a 1 n . . . a i 1 ··· a ij ··· a in . . . a n 1 ··· . . . a nj . . . ··· a nn
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Chapter 2. Determinants Math1111 Determinants Minor Definition (Minor) Let A = a 11 ··· . . . a 1 j . . . ··· a 1 n . . . a i 1 ··· a ij ··· a in . . . a n 1 ··· . . . a nj . . . ··· a nn
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Chapter 2. Determinants Math1111 Determinants Minor Definition (Minor) Let A = a 11 ··· . . . a 1 j . . . ··· a 1 n . . . a i 1 ··· a ij ··· a in . . . a n 1 ··· . . . a nj . . . ··· a nn The minor of a ij is M ij = det
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Chapter 2. Determinants Math1111 Determinants Minor Definition (Minor) Let A = a 11 ··· . . . a 1 j . . . ··· a 1 n . . . a i 1 ··· a ij ··· a in . . . a n 1 ··· . . . a nj . . . ··· a nn The minor of a ij is M ij = det
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Chapter 2. Determinants Math1111 Determinants Minor Definition (Minor) Let A = a 11 ··· .
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