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Chapter2 - Chapter 2 Determinants and Eigenvalues MAT188H1F...

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Chapter 2: Determinants and Eigenvalues MAT188H1F Lec0106 Burbulla Chapter 2 Lecture Notes Fall 2010 Chapter 2 Lecture Notes MAT188H1F Lec0106 Burbulla Chapter 2: Determinants and Eigenvalues Chapter 2: Determinants and Eigenvalues 2.1: Cofactor Expansions 2.2: Determinants and Inverses 2.3: Diagonalization and Eigenvalues 2.5, 2.6 and 2.7: Three Extra Topics 2.8: Systems of Differential Equations Chapter 2 Lecture Notes MAT188H1F Lec0106 Burbulla
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Chapter 2: Determinants and Eigenvalues 2.1: Cofactor Expansions 2.2: Determinants and Inverses 2.3: Diagonalization and Eigenvalues 2.5, 2.6 and 2.7: Three Extra Topics 2.8: Systems of Differential Equations The Determinant of a 2 × 2 Matrix The determinant of the matrix A = a b c d is ad - bc and we write det A = ad - bc . So det 3 2 5 7 = 21 - 10 = 11 . and det - 1 6 4 8 = - 8 - 24 = - 32 . The formula for the inverse of a 2 × 2 matrix involves determinants: A - 1 = a b c d - 1 = 1 ad - bc d - b - c a = 1 det A d - b - c a . Chapter 2 Lecture Notes MAT188H1F Lec0106 Burbulla Chapter 2: Determinants and Eigenvalues 2.1: Cofactor Expansions 2.2: Determinants and Inverses 2.3: Diagonalization and Eigenvalues 2.5, 2.6 and 2.7: Three Extra Topics 2.8: Systems of Differential Equations 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 the following expression is not zero: 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 . This expression is defined to be the determinant of A : 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 . Chapter 2 Lecture Notes MAT188H1F Lec0106 Burbulla
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Chapter 2: Determinants and Eigenvalues 2.1: Cofactor Expansions 2.2: Determinants and Inverses 2.3: Diagonalization and Eigenvalues 2.5, 2.6 and 2.7: Three Extra Topics 2.8: Systems of Differential Equations 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 Lec0106 Burbulla Chapter 2: Determinants and Eigenvalues 2.1: Cofactor Expansions 2.2: Determinants and Inverses 2.3: Diagonalization and Eigenvalues 2.5, 2.6 and 2.7: Three Extra Topics 2.8: Systems of Differential Equations Example 1 det 2 1 - 1 1 1 1 - 3 2 4 = 8 + ( - 3) + ( - 2) - (3) - 4 - 4 = - 8 .
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