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CHAPTER 1 DETERMINANTS 1

1.1 2×2 Determinant and 3×3 Determinant 1

1.1.1 2×2 Determinant 1

1.1.2 3×3 Determinant 2

1.2 n×n Determinants 2

1.2.1 Permutations and the Number of the Inversions 2

1.2.2 n×n Determinants 4

1.3 The Properties of the Determinants 6

1.4 LAPLACE Expansion of Determinants 12

1.4.1 LAPLACE Expansion of Determinants along a Row or a Column 12

1.4.2 LAPLACE Expansion of Determinants along k Rows or k Columns 17

1.5 Cramer's Rule 17

Exercises 1 (A) 21

Exercises 1 (B) 26

CHAPTER 2 MATRICES 28

2.1 Concepts of Matrices 28

2.2 Operations of Matrices 31

2.2.1 Addtion Operation of Matrices as well as the Multiplication of Scalar and Matrix 31

2.2.2 Multiplications of Matrices 33

2.2.3 Matrix Transpose 38

2.2.4 Power of Square Matrices 39

2.3 Some Special Matrices 39

2.3.1 Diagonal Matrices 39

2.3.2 Scalar Matrices 40

2.3.3 Identity Matrices 40

2.3.4 Lower-Triangular Matrices 40

2.3.5 Upper-Triangular Matrices 41

2.3.6 Symmetric Matrices 41

2.4 Block-Divided Matrices 42

2.5 Invertible Matrices 46

2.6 Elementary Operations on Matrices 50

2.7 Rank of Matrices 57

Exercises 2 (A) 60

Exercises 2 (B) 66

CHAPTER 3 LINEAR EQUATIONS 69

3.1 Gaussian Elimination for Solving Linear Equations 69

3.2 Vector Space of n Dimensions 79

3.2.1 Vectors and Their Linear Operations 79

3.3 Linear Relations of Vectors 80

3.3.1 Linear Combinations of Vectors 80

3.3.2 Linear Dependence and Linear Independence of Vectors 82

3.3.3 Linear Dependence and Linear Combination of Vectors 85

3.3.4 Rank of Vectors 88

3.4 Solution Structures of Linear Equations 92

3.4.1 Solution Structures of Homogeneous Linear Equations 92

3.4.2 Solution Structures of Nonhomogeneous Linear Equations 96

Exercises 3 (A) 98

Exercises 3 (B) 101

CHAPTER 4 EIGENVALUES OF MATRICES 104

4.1 Eigenvalues and Eigenvectors of Matrices 104

4.1.1 Eigenvalues of Matrices 104

4.1.2 Fundamental Properties of the Eigenvalues and the Eigenvectors of Matrices 106

4.2 Similar Matrices 107

4.2.1 Similar Matrices and Their Properties 107

4.2.2 Conditions for an n×n Matrix A to be Simiar to a Diagonal Matrix 108

4.2.3 The Jordan Canonical Form of Matrices 110

4.3 Eigenvalues and Eigenvectors of Real Symmetric Matrices 112

4.3.1 Inner Products of Vectors 112

4.3.2 Orthogonal Vectors 113

4.3.3 Orthogonal Matrices 114

4.3.4 Eigenvalues and Eigenvectors of Real Symmetric Matrices 115

Exercises 4 118

CHAPTER 5 QUADRATIC FORMS 120

5.1 Quadratic Forms and Symmetric Matrices 120

5.1.1 Quadratic Forms and the Related Symmetric Matrices 120

5.1.2 Linear Transformations 121

5.2 Normalized Forms Quadratic Forms and Symmetric Matrices 124

5.2.1 Normalizing Quadratic Forms and the Related Symmetric Matrices by Completing Squares 124

5.2.2 Elementary Operations Used in Deducing Quadratic Forms to Normalized Ones 127

5.2.3 Normalizing Quadratic Forms with Orthogonal Linear Transformations 128

5.2.4 Normalized Forms of Quadratic Forms and the Corresponding Matrices 130

5.3 Quadratic Forms and Definite Matrices 131

5.4 An Application of Definite Quadratic Forms 137

Exercises 5 (A) 139

Exercises 5 (B) 140