section21 - Chapter 2 MAT188H1F Lec03 Burbulla Chapter 2...

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Chapter 2 MAT188H1F Lec03 Burbulla Chapter 2 Lecture Notes Fall 2007 Chapter 2 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 2 Chapter 2 2.1: Cofactor Expansions Chapter 2 Lecture Notes MAT188H1F Lec03 Burbulla
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Chapter 2 2.1: Cofactor Expansions The Determinant of a 2 × 2 Matrix The determinant of the matrix A = ± a b c d ² is ad - bc and we write det A = ad - bc . So det ± 3 2 5 7 ² = 21 - 10 = 11 . and det ± - 1 6 4 8 ² = - 8 - 24 = - 32 . The formula for the inverse of a 2 × 2 matrix involves determinants: A - 1 = ± a b c d ² - 1 = 1 ad - bc ± d - b - c a ² = 1 det A ± d - b - c a ² . Chapter 2 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 2 2.1: Cofactor Expansions The Determinant of a 3 × 3 Matrix Let A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 There are only nine entries here, so it should be possible to row-reduce A I and find which conditions the entries a ij must satisfy for there to be three leading 1’s in the reduced row-echelon form of A . Try it! It’s very tedious and messy, but you would find A is invertible if and only if a 11 a 22 a 33 + a 12 a 23 a 31 + a 13 a 21 a 32 - a 31 a 22 a 13 - a 32 a 23 a 11 - a 33 a 21 a 12 is not zero. This expression is defined to be the determinant of the 3 × 3 matrix A , and we write det A = a 11 a 22 a 33 + a 12 a 23 a 31 + a 13 a 21 a 32 - a 31 a 22 a 13 - a 32 a 23 a 11 - a 33 a 21 a 12 . Memorize it! Chapter 2 Lecture Notes MAT188H1F Lec03 Burbulla
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2.1: Cofactor Expansions Ways to Remember the Determinant of a 3 × 3 Matrix, A Consider the 3 × 5 array, obtained from A by repeating the first and second columns: a 11 a 12 a 13 a 11 a 12 a 21 a 22 a 23 a 21 a 22 a 31 a 32 a 33 a 31 a 32 In this array there are three diagonals that go from top-left to bottom-right, and there are three diagonals that go from bottom-left to top-right. Add the products of the terms on the first three diagonals, and subtract the products of the terms of the last three diagonals. That will give you det A . This is somewhat analogous to the formula for the determinant of a 2 × 2 matrix, but no such schemes will work for any matrices larger than 3 × 3 . Chapter 2 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 2 2.1: Cofactor Expansions Example 1 det 2 1 - 1 1 1 1 - 3 2 4 = 8 + ( - 3) + ( - 2) - (3) - 4 - 4 = - 8 . det 1 1 - 1 1 - 1 1 - 1 1 1 = - 1 + ( - 1) + ( - 1) - ( - 1) - 1 - 1 = - 4 det 2 1 - 11 0 10 7 0 0 - 4 = - 80 + 0 + 0 - 0 - 0 - 0 = - 80 . det 1 1 1 2 1 - 1 3 2 0 = 0 + ( - 3) + 4 - 3 - ( - 2) - 0 = 0 . Chapter 2 Lecture Notes
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section21 - Chapter 2 MAT188H1F Lec03 Burbulla Chapter 2...

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