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Chapter 3 Determinants

Chapter 3 Determinants. 3.1 The Determinant of a Matrix 3.2 Evaluation of a Determinant using Elementary Operations 3.3 Properties of Determinants 3.4 Introduction to Eigenvalues 3.5 Application of Determinants.

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Chapter 3 Determinants

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  1. Chapter 3Determinants 3.1 The Determinant of a Matrix 3.2 Evaluation of a Determinant using Elementary Operations 3.3 Properties of Determinants 3.4 Introduction to Eigenvalues 3.5 Application of Determinants Elementary Linear Algebra 投影片設計編製者 R. Larsen et al. (6 Edition) 淡江大學 電機系 翁慶昌 教授

  2. 3.1 The Determinant of a Matrix • the determinant of a 2 × 2 matrix: • Note: Elementary Linear Algebra: Section 3.1, p.123

  3. Ex. 1: (The determinant of a matrix of order 2) • Note: The determinant of a matrix can be positive, zero, or negative. Elementary Linear Algebra: Section 3.1, p.123

  4. Minor of the entry : The determinant of the matrix determined by deleting the ith row and jth column of A • Cofactor of : Elementary Linear Algebra: Section 3.1, p.124

  5. Ex: Elementary Linear Algebra: Section 3.1, p.124

  6. Notes: Sign pattern for cofactors 3 × 3 matrix 4 × 4 matrix n ×n matrix • Notes: Odd positions (where i+j is odd) have negative signs, and even positions (where i+j is even) have positive signs. Elementary Linear Algebra: Section 3.1, p.124

  7. Ex 2: Find all the minors and cofactors of A. Sol: (1) All the minors of A. Elementary Linear Algebra: Section 3.1, p.125

  8. Sol: (2) All the cofactors of A. Elementary Linear Algebra: Section 3.1, p.125

  9. Thm 3.1: (Expansion by cofactors) Let A isasquare matrix of order n. Then the determinant of A is given by (Cofactor expansion along the i-th row, i=1, 2,…, n) or (Cofactor expansion along the j-th row, j=1, 2,…, n ) Elementary Linear Algebra: Section 3.1, p.126

  10. Ex: The determinant of a matrix of order 3 Elementary Linear Algebra: Section 3.1, Addition

  11. Ex 3: The determinant of a matrix of order 3 Sol: Elementary Linear Algebra: Section 3.1, p.126

  12. Ex 5: (The determinant of a matrix of order 3) Sol: Elementary Linear Algebra: Section 3.1, p.128

  13. Notes: The row (or column) containing the most zeros is the best choice for expansion by cofactors . • Ex 4: (The determinant of a matrix of order 4) Elementary Linear Algebra: Section 3.1, p.127

  14. Sol: Elementary Linear Algebra: Section 3.1, p.127

  15. The determinant of a matrix of order 3: Subtract these three products. Add these three products. Elementary Linear Algebra: Section 3.1, p.127

  16. Ex 5: –4 0 6 0 –12 16 Elementary Linear Algebra: Section 3.1, p.128

  17. Upper triangular matrix: All the entries below the main diagonal are zeros. • Lower triangular matrix: All the entries above the main diagonal are zeros. • Diagonal matrix: All the entries above and below the main diagonal are zeros. • Note: A matrix that is both upper and lower triangular is called diagonal. Elementary Linear Algebra: Section 3.1, p.128

  18. Ex: diagonal lower triangular upper triangular Elementary Linear Algebra: Section 3.1, p.128

  19. Thm 3.2: (Determinant of a Triangular Matrix) If A is an n × n triangular matrix (upper triangular, lower triangular, or diagonal), then its determinant is the product of the entries on the main diagonal. That is Elementary Linear Algebra: Section 3.1, p.129

  20. Ex 6: Find the determinants of the following triangular matrices. (b) (a) Sol: |A| = (2)(–2)(1)(3) = –12 (a) |B| = (–1)(3)(2)(4)(–2) = 48 (b) Elementary Linear Algebra: Section 3.1, p.129

  21. Keywords in Section 3.1: • determinant : 行列式 • minor : 子行列式 • cofactor : 餘因子 • expansion by cofactors : 餘因子展開 • upper triangular matrix: 上三角矩陣 • lower triangular matrix: 下三角矩陣 • diagonal matrix: 對角矩陣

  22. 3.2 Evaluation of a determinant using elementary operations • Thm 3.3: (Elementary row operations and determinants) Let A and B be square matrices. Elementary Linear Algebra: Section 3.2, p.134

  23. Ex: Elementary Linear Algebra: Section 3.2, Addition

  24. Notes: Elementary Linear Algebra: Section 3.2, pp.134-135

  25. Sol: Note: A row-echelon form of a square matrix is always upper triangular. • Ex 2: (Evaluation a determinant using elementary row operations) Elementary Linear Algebra: Section 3.2, pp.134-135

  26. Elementary Linear Algebra: Section 3.2, pp.134-135

  27. Notes: Elementary Linear Algebra: Section 3.2, Addition

  28. Determinants and elementary column operations • Thm: (Elementary column operations and determinants) Let A and B be square matrices. Elementary Linear Algebra: Section 3.2, p.135

  29. Ex: Elementary Linear Algebra: Section 3.2, p.135

  30. Thm 3.4: (Conditions that yield a zero determinant) If A is a square matrix and any of the following conditions is true, then det (A) = 0. (a) An entire row (or an entire column) consists of zeros. (b) Two rows (or two columns) are equal. (c) One row (or column) is a multiple of another row (or column). Elementary Linear Algebra: Section 3.2, p.136

  31. Ex: Elementary Linear Algebra: Section 3.2, Addition

  32. Note: Elementary Linear Algebra: Section 3.2, p.138

  33. Ex 5: (Evaluating a determinant) Sol: Elementary Linear Algebra: Section 3.2, p.138

  34. Ex 6: (Evaluating a determinant) Sol: Elementary Linear Algebra: Section 3.2, p.139

  35. Elementary Linear Algebra: Section 3.2, p.140

  36. 3.3 Properties of Determinants • Notes: • Thm 3.5: (Determinant of a matrix product) det (AB) = det (A) det (B) (1) det (EA) = det (E) det (A) (2) (3) Elementary Linear Algebra: Section 3.3, p.143

  37. Ex 1: (The determinant of a matrix product) Find |A|, |B|, and |AB| Sol: Elementary Linear Algebra: Section 3.3, p.143

  38. Check: |AB| = |A| |B| Elementary Linear Algebra: Section 3.3, p.143

  39. Ex 2: • Thm 3.6: (Determinant of a scalar multiple of a matrix) If A is an n × n matrix and c is a scalar, then det (cA) = cn det (A) Find |A|. Sol: Elementary Linear Algebra: Section 3.3, p.144

  40. Thm 3.7: (Determinant of an invertible matrix) • Ex 3: (Classifying square matrices as singular or nonsingular) A square matrix A is invertible (nonsingular) if and only if det (A)  0 Sol: A has no inverse (it is singular). B has an inverse (it is nonsingular). Elementary Linear Algebra: Section 3.3, p.145

  41. Ex 4: • Thm 3.8: (Determinant of an inverse matrix) • Thm 3.9: (Determinant of a transpose) (a) (b) Sol: Elementary Linear Algebra: Section 3.3, pp.146-148

  42. If A is an n × n matrix, then the following statements are equivalent. • Equivalent conditions for a nonsingular matrix: (1) A is invertible. (2) Ax = b has a unique solution for every n × 1 matrix b. (3) Ax = 0 has only the trivial solution. (4) A is row-equivalent to In (5) A can be written as the product of elementary matrices. (6) det (A)  0 Elementary Linear Algebra: Section 3.3, p.147

  43. Ex 5: Which of the following system has a unique solution? (a) (b) Elementary Linear Algebra: Section 3.3, p.148

  44. Sol: (a) This system does not have a unique solution. (b) This system has a unique solution. Elementary Linear Algebra: Section 3.3, p.148

  45. Eigenvalue and eigenvector: A：an nn matrix ：a scalar x： a n1nonzero column matrix Eigenvalue Eigenvector (The fundamental equation for the eigenvalue problem) 3.4 Introduction to Eigenvalues • Eigenvalue problem: If A is an nn matrix, do there exist n1 nonzero matrices x such that Ax is a scalar multiple of x？ Elementary Linear Algebra: Section 3.4, p.152

  46. Eigenvalue Eigenvalue Eigenvector Eigenvector • Ex 1: (Verifying eigenvalues and eigenvectors) Elementary Linear Algebra: Section 3.4, p.152

  47. Note: (homogeneous system) If has nonzero solutions iff . • Characteristic equation of AMnn: • Question: Given an nn matrix A, how can you find the eigenvalues and corresponding eigenvectors? Elementary Linear Algebra: section 3.4, p.153

  48. Sol: Characteristic equation: Eigenvalues: • Ex 2: (Finding eigenvalues and eigenvectors) Elementary Linear Algebra: Section 3.4, p.153

  49. Elementary Linear Algebra: Section 3.4, p.153

  50. Ex 3: (Finding eigenvalues and eigenvectors) Sol: Characteristic equation: Elementary Linear Algebra: Section 3.4, p.154

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