Chapter2 - Chapter 2: Determinants and Eigenvalues...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 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 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 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 ....
View Full Document

This note was uploaded on 12/23/2010 for the course MAT mat 188 taught by Professor Dietrichburbulla during the Spring '10 term at University of Toronto- Toronto.

Page1 / 50

Chapter2 - Chapter 2: Determinants and Eigenvalues...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online