Chapter 3 Determinants

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 投影片設計製作者 R. Larsen et al. (5 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:

3. Ex. 1: (The determinant of a matrix of order 2) • Note: The determinant of a matrix can be positive, zero, or negative.

4. Minor of the entry : • Cofactor of :

5. Ex: • Notes: Sign pattern for cofactors

6. 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 )

7. Ex: The determinant of a matrix of order 3

8. Ex 5: (The determinant of a matrix of order 3) Sol:

9. Ex 4: (The determinant of a matrix of order 4)

10. Sol:

11. The determinant of a matrix of order 3: Subtract these three products. Add these three products.

12. Ex 5: –4 0 6 0 –12 16

13. Upper triangular matrix: All the entries below the main diagonal are zeros. • Lower triangular matrix: All the entries below the main diagonal are zeros. • Diagonal matrix: All the entries above and below the main diagonal are zeros.

14. Ex: diagonal lower triangular upper triangular

15. Thm 3.2: (Determinant of a Triangular Matrix) If A is an nxn triangular matrix (upper triangular, lower triangular, or diagonal), then its determinant is the product of the entries on the main diagonal. That is

16. 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)

17. 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

18. 3.2 Evaluation of a determinant using elementary operations • Thm 3.3: (Elementary row operations and determinants) Let A and B be square matrices.

19. Ex:

20. Sol: Note: A row-echelon form of a square matrix is always upper triangular. • Ex 2: (Evaluation a determinant using elementary row operations)

21. Notes:

22. Notes:

23. 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).

24. Ex:

25. Note:

26. Ex 5: (Evaluating a determinant) Sol:

27. Ex 6: (Evaluating a determinant) Sol:

29. 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)

30. Ex 1: (The determinant of a matrix product) Find |A|, |B|, and |AB| Sol:

31. Check: |AB| = |A| |B|

32. 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:

33. 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 inverse (it is nonsingular).

34. Ex 4: • Thm 3.8: (Determinant of an inverse matrix) • Thm 3.9: (Determinant of a transpose) (a) (b) Sol:

35. 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

36. Ex 5: Which of the following system has a unique solution? (a) (b)

37. Sol: (a) This system does not have a unique solution. (b) This system has a unique solution.

38. 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？

39. Eigenvalue Eigenvalue Eigenvector Eigenvector • Ex 1: (Verifying eigenvalues and eigenvectors)

40. 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?

41. Sol: Characteristic equation: Eigenvalue: • Ex 2: (Finding eigenvalues and eigenvectors)

43. Ex 3: (Finding eigenvalues and eigenvectors) Sol: Characteristic equation:

46. 3.5 Applications of Determinants • Matrix of cofactors of A: • Adjoint matrix of A:

47. Thm 3.10: (The inverse of a matrix given by its adjoint) If A is an n × n invertible matrix, then • Ex:

48. Ex 2: (a) Find the adjoint of A. (b) Use the adjoint of A to find Sol:

49. adjoint matrix of A inverse matrix of A • Check: cofactor matrix of A

50. Thm 3.11: (Cramer’s Rule) (this system has a unique solution)