Lect8 - Chapter 2. Determinants Math1111 Determinants...

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Unformatted text preview: Chapter 2. Determinants Math1111 Determinants Motivation Motivation : 1 × 1 case : When is A = ( a ) invertible? Ans: a 6 = . 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 = Chapter 2. Determinants Math1111 Determinants Motivation Motivation : 1 × 1 case : When is A = ( a ) invertible? Ans: a 6 = . 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 = Determinant is a number associated to a square matrix, whose value tells whether the matrix is invertible. Chapter 2. Determinants Math1111 Determinants Motivation Motivation : 1 × 1 case : When is A = ( a ) invertible? Ans: a 6 = . 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 = Determinant is a number associated to a square matrix, whose value tells whether the matrix is invertible. Notation : det ( A ) , det A , | A | Chapter 2. Determinants Math1111 Determinants Definition (order 1) Determinant is defined inductively. Det of order 1 → Det of order 2 → Det of order 3 → ··· 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 . 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 . Chapter 2. Determinants Math1111 Determinants Minor Definition (Minor) Let A = a 1 1 ··· . . . a 1 j . . . ··· a 1 n . . . a i 1 ··· a i j ··· a i n . . . a n 1 ··· . . . a n j . . . ··· a n n Chapter 2. Determinants Math1111 Determinants Minor Definition (Minor) Let A = a 1 1 ··· . . . a 1 j . . . ··· a 1 n . . . a i 1 ··· a i j ··· a i n . . . a n 1 ··· . . . a n j . . . ··· a n n Chapter 2. Determinants Math1111 Determinants Minor Definition (Minor) Let A = a 1 1 ··· . . . a 1 j . . . ··· a 1 n . . . a i 1 ··· a i j ··· a i n . . . a n 1 ··· . . . a n j ....
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Lect8 - Chapter 2. Determinants Math1111 Determinants...

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