1. 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
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
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 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 is a square 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. The row (or column) containing the most zeros is the best choice
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 6 16
–12
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 upper triangular lower 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)
Sol:
(a)
|A| = (2)(–2)(1)(3) = –12
(b)
|B| = (–1)(3)(2)(4)(–2) = 48
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

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

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: Cofactor Expansion Row Reduction Order n 3 5 9 10 119 205 30 45
Additions
Multiplications 3 5 9 10
119
205 30 45
3,628,799
6,235,300
285
339
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
Thm 3.5: (Determinant of a matrix product)
det (AB) = det (A) det (B)
Notes:
(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. 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)
Ex 2:
Find |A|.
Sol:
Elementary Linear Algebra: Section 3.3, p.144

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

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

42. Equivalent conditions for a nonsingular matrix:
If A is an n × n matrix, then the following statements are equivalent.
(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?
(b)
Elementary Linear Algebra: Section 3.3, p.148

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

45. 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？
Eigenvalue and eigenvector:
A：an nn matrix
：a scalar
x： a n1 nonzero column matrix
Eigenvalue
Eigenvector
(The fundamental equation for
the eigenvalue problem)
Elementary Linear Algebra: Section 3.4, p.152

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

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

48. Ex 2: (Finding eigenvalues and eigenvectors)
Sol: Characteristic equation:
Eigenvalues:
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

51. Elementary Linear Algebra: Section 3.4, p.155

52. Elementary Linear Algebra: Section 3.4, p.156

53. 3.5 Applications of Determinants
Matrix of cofactors of A:
Adjoint matrix of A:
Elementary Linear Algebra: Section 3.5, p.158

54. Thm 3.10: (The inverse of a matrix given by its adjoint)
If A is an n × n invertible matrix, then
Ex:
Elementary Linear Algebra: Section 3.5, p.159

55. (a) Find the adjoint of A.
Ex 1 & Ex 2:
(a) Find the adjoint of A.
(b) Use the adjoint of A to find
Sol:
Elementary Linear Algebra: Section 3.5, p.160

56. adjoint matrix of A cofactor matrix of A inverse matrix of A Check:
Elementary Linear Algebra: Section 3.5, p.160

57. (this system has a unique solution)
Thm 3.11: (Cramer’s Rule)
(this system has a unique solution)
Elementary Linear Algebra: Section 3.5, p.163

58. ( i.e. )
Elementary Linear Algebra: Section 3.5, p.163

59. Pf:
A x = b,
Elementary Linear Algebra: Section 3.5, p.163

60. Elementary Linear Algebra: Section 3.5, p.163

61. Ex 4: Use Cramer’s rule to solve the system of linear equations.
Elementary Linear Algebra: Section 3.5, p.163

62. Keywords in Section 3.5: matrix of cofactors : 餘因子矩陣
adjoint matrix : 伴隨矩陣
Cramer’s rule : Cramer 法則