Chapter 3: Matrices and Determinants

3.1: Introduction

1. Introduction to Matrix
2. More on Introducing Matrix
3. Problem 1: Introduction to Matrix
4. Order of Matrix
5. Problem 1: Order of Matrix
6. Equal Matrices
7. Problem 1: Equal Matrices
8. Problem 2: Equal Matrices
9. Row Matrix and Column Matrix
10. Problem 1: Row Matrix and Column Matrix
11. Square Matrix and Rectangular Matrix
12. Problem 1: Square Matrix and Rectangular Matrix
13. Null Matrix or Zero Matrix
14. Problem 1: Null Matrix or Zero Matrix
15. Practical Application of Matrices
16. Transpose of a Matrix
17. Problem 1: Transpose of a Matrix
18. Negative of Matrix
19. Problem 1: Negative of Matrix
20. Symmetric and Skew-Symmetric Matrices
21. Problem 1: Symmetric and Skew-Symmetric Matrices
22. Problem2-Symmetric & Skew-Symmetric Matrices
23. Hermitian and Skew Hermitian Matrix
24. Problem-Hermitian and Skew Hermitian Matrix
25. Triangular Matrices
26. Problem-Triangular Matrices
27. Diagonal Matrix
28. Problem 1: Diagonal Matrix
29. Scalar Matrix
30. Problem 1: Scalar Matrix
31. Identity Matrix
32. Problem 1: Identity Matrix
33. Adjoint of a Matrix
34. Problem 1: Adjoint of a Matrix
35. Addition of Matrices
36. Problem 1: Addition of Matrices
37. Subtraction of Matrices
38. Problem 1: Subtraction of Matrices
39. Multiplication of Matrix by a Real Number
40. Problem 1: Multiplication of Matrix by a Real Number
41. Multiplicative Identity of a Matrix
42. Problem 1: Multiplicative Identity of a Matrix
43. Properties of Transposed Matrices
44. Problem-Properties of Transposed Matrices
45. Multiplication of Matrices
46. More on Multiplication of Matrices
47. Problem 1: Multiplication of Matrices
48. Problem 2: Multiplication of Matrices
49. Problem3-Multiplication of Matrices

3.2: Determinant of a 2x2 Matrix

1. Determinant of 2-by-2 Matrix
2. Problem1-Singular and Non-Singular Matrix
3. Problem2-Singular and Non-Singular Matrix
4. Singular and Non-singular Matrix
5. Problem 1: Singular and Non-singular Matrix
6. Adjoint of a Matrix
7. Problem 1: Adjoint of a Matrix
8. Multiplicative Inverse of a Non-Singular Matrix
9. Problem 1: Multiplicative Inverse of a Non-Singular Matrix
10. Inverse of Matrix using Adjoint
11. More on Matrix Inverse Using Adjoint
12. Problem 1: Inverse of Matrix using Adjoint
13. Problem2-Matrix Inverse Using Adjoint
14. Problem3-Matrix Inverse Using Adjoint
15. Problem4-Matrix Inverse Using Adjoint

3.3: Solution of Simultaneous Linear Equations By Using Matrices

1. Solving Simultaneous Equations-Inversion Method
2. Use of Matrices in Solving Everyday Life Problems

3.4: Field

1. Introducing Field
2. Problem-Introducing Field

3.5: Properties of Matrix Addition, Scalar Multiplication and Matrix Multiplication

1. Commutative Law under Addition for Matrices
2. Problem 1: Commutative Law under Addition for Matrices
3. Associaitve Law under Addition for Matrices
4. Problem 1: Associaitve Law under Addition for Matrices
5. Additive Identity of a Matrix
6. Problem 1: Additive Identity of a Matrix
7. Additive Inverse of a Matrix
8. Problem 1: Additive Inverse of a Matrix
9. Multiplication of Matrices
10. Problem 1: Multiplication of Matrices
11. Problem 2: Multiplication of Matrices
12. Associative law under Multiplication of matrices
13. Problem 1: Associative law under Multiplication of matrices
14. Distributive Law of Multiplication over Addition for Matrices
15. Problem 1: Distributive Law of Multiplication over Addition for Matri
16. Commutative Law of Multiplication of Matrices
17. Problem 1: Commutative Law of Multiplication of Matrices
18. Prove That c(AB) = (cA)B = A(cB)
19. Problem-Prove That c(AB) = (cA)B = A(cB)
20. Properties of Scalar Multiplication

3.6: Determinants

1. Minor of an Element of a Matrix or Its Determinant
2. Problem-Minor of an Element of a Matrix or Its Determinant
3. Cofactor of an Element of a Matrix
4. Problem-Cofactor of an Element of a Matrix
5. Determinant of a Sqaure Matrix of Order 3 or greater
6. Problem-Determinant of a Sqaure Matrix of Order 3 or greater

3.7: Properties of Determinants Which Help in Their Evaluation

1. Properties of Determinants
2. More on Properties of Determinants
3. Problem 1-Properties of Determinants
4. Problem 2-Properties of Determinants
5. Problem 3-Properties of Determinants
6. Problem 4-Properties of Determinants
7. Properties of Determinants of Order Three
8. Problem-Properties of Determinants of Order Three
9. Alternate Method For Expanding a Third Order Determinant
10. Problem-Alternate Method For Expanding a Third Order Determinant

3.8: Adjoint and Inverse of a Square Matrix of Order n = 3 or n > 3

1. Inverse of Matrix using Adjoint
2. More on Matrix Inverse Using Adjoint
3. Problem 1: Inverse of Matrix using Adjoint
4. Problem2-Matrix Inverse Using Adjoint
5. Adjoint of a Square Matrix of Order n = 3 or n > 3
6. Problem-Adjoint of a Square Matrix of Order n = 3 or n > 3
7. Property for whole transpose of product of Matrices
8. Property for Whole Inverse of Product of Matrices
9. More on Property for Whole Inverse of Product of Matrices
10. Problem 1: Property for whole transpose of product of Matrices

3.9: Elementary Row and Column Operations on a Matrix

1. Deriving a Method For Determining Inverses
2. Problem-Deriving a Method For Determining Inverses
3. Inverse of Matrix by Row Operation
4. Problem-Inverse of Matrix by Row Operation
5. Problem-More on Inverse of Matrix by Row Operation
6. Inverse of Matrix by Column Operation
7. Problem-Inverse of Matrix by Column Operation
8. Problem-More on Inverse of Matrix by Column Operation
9. Triangular Matrices
10. Problem-Triangular Matrices
11. Symmetric and Skew-Symmetric Matrices
12. Problem 1: Symmetric and Skew-Symmetric Matrices
13. Hermitian and Skew Hermitian Matrix
14. Problem-Hermitian and Skew Hermitian Matrix

3.10: Echelon and Reduced Echelon Forms of Matrices

1. Echelon and Reduced Echelon Form of Matrix
2. Problem-Echelon and Reduced Echelon Form of Matrix
3. Echelon Form of Matrix
4. Problem-Echelon Form of Matrix
5. Reduced Echelon Form of Matrix
6. Problem-Reduced Echelon Form of Matrix
7. Rank of a Matrix
8. Problem-Rank of a Matrix
9. Invertible Matrix
10. Problem-Invertible Matrix
11. Theorems on Invertible Matrix
12. More Theorems on Invertible Matrix

3.11: Systems of Linear Equations

1. Solution of System of Linear Equations by Row Operation
2. Problem-Solution of System of Linear Equations by Row Operation
3. Problem-More on Solution of System of Linear Equations by Row Operati
4. Solution of Homogeneous Linear Equations
5. More on Solution of Homogeneous Linear Equations
6. Problem-Solution of Homogeneous Linear Equations
7. More on Problem-Solution of Homogeneous Linear Equations

3.12: Cramer's Rule

1. System of Linear Equations
2. More on System of Linear Equations
3. Problem-System of Linear Equations
4. Consistency and Inconsistency of a System
5. Problem-Consistency and Inconsistency of a System
6. Solving a System of Three Non-Homogeneous Linear Equations
7. Problem-Solving a System of Three Non-Homogeneous Linear Equations
8. Solving Simultaneous Equations-Cramers Rule
9. More on Use of Matrices Solving Everyday Life Problems
10. More on Solving Simultaneous Equations-Cramers Rule
11. Problem-Solving Simultaneous Equations-Cramers Rule