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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 6 = 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 6 = 0

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Chapter 2. Determinants Math1111 Determinants Motivation Motivation: 1 × 1 case : When is A = ( a ) invertible? Ans: a 6 = 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 6 = 0 Determinant is a number associated to a square matrix, whose value tells whether the matrix is invertible.
Chapter 2. Determinants Math1111 Determinants References Leon (7th edition) Chapter 2. Sections 3 (8th edition) Chapter 2. Sections 2.1-2.3 DeFranza Chapter 1. Sections 1.6 Nicholson (1st edition) Chapter 2. Sections 2.1-2.2 (2nd edition) Chapter 2. Sections 2.1-2.2

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Chapter 2. Determinants Math1111 Determinants Definition (order 1) Notation: det ( A ) , det A , | A | Determinant is defined inductively. Det of order 1 Det of order 2 Det of order 3 → ···
Chapter 2. Determinants Math1111 Determinants Definition (order 1) Notation: det ( A ) , det A , | A | 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) Notation: det ( A ) , det A , | A | 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 .
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
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
Chapter 2. Determinants Math1111 Determinants Minor Definition (Minor) Let A = a 11 ··· .

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Lect8 - Chapter 2 Determinants Math1111 Determinants...

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