# Book:A.C. Aitken/Determinants and Matrices/Eighth Edition

## A.C. Aitken: Determinants and Matrices (8th Edition)

Published $\text {1954}$, Oliver & Boyd.

### Contents

Chapter $\text I$: Definitions and Fundamental Operations of Matrices
1. Introductory
2. Linear Equations and Transformations
3. The Notation of Matrices
4. Matrices, Row and Column Vectors, Scalars
5. The Operations of Matrix Algebra
6. Matrix Pre- and Postmultiplication
7. Product of Three or More Matrices
8. Transposition of Rows and Columns
9. Transpose of a Product: Reversal Rule
10. Algebraic Expressions in Matrix Notation
11. Partitioned Matrices and Multiplication
Chapter $\text {II}$: Definition and Properties of Determinants
12. The Solution of Simultaneous Equations
13. Salient Properties of Eliminants
14. Inversions and Class of Permutations
15. Definition and Notation of Determinant
16. Identity of Class of Conjugate Permutations
17. Elementary Properties of Determinants
18. Primeness of a Determinant
19. Various Expansions of a Determinant
20. Pivotal Evaluation of Determinants
Chapter $\text {III}$: Adjugate and Reciprocal Matrix: Solution of Simultaneous Equations: Rank and Linear Dependence
21. The Adjugate Matrix of a Square Matrix
22. Solution of Equations in Nonsingular Case
23. Reversal Rule for Reciprocal of Product
24. Orthogonal and Unitary Matrices
25. Solution of Homogeneous Equations
26. Rank and Nullity of Matrix
27. Linear Dependence
28. Conditions for Solution of Homogeneous Equations
29. Reduction of Matrix to Equivalent Form
30. Solution of Non-Honogeneous Equations
Chapter $\text {IV}$: Cauchy and Laplace Expansions: Multiplication Theorems
31. Expansion by Elements of Row and Column
32. Complementary Minors and Cofactors
33. Laplacian Expansion of a Determinant
34. Multiplication of Determinants
35. Extended Laplace and Cauchy Expansions
36. Determinant of Product of Rectangular Matrices
37. Expansion by Diagonal Elements: Normal Form
Chapter $\text V$: Compound Matrices and Determinants: Dual Theorems
38. Compound and Adjugate Compound Matrices
39. Binet-Cauchy Theorem on Product of Compounds
40. Reciprocal of Nonsingular Compound Matrix
41. Rank Expressed by Compound Matrices
42. Jacobi's Theorem on Minors of Adjugate
43. Franke's Theorem on Minors of Compound
44. Hybrid Compounds of Bazin and Reiss
45. Complementary and Extensional Identities
46. Schweinsian Expansions of Determinant Quotients
Chapter $\text {VI}$: Special Determinants: Alternant, Persymmetric, Bigradient, Centrosymmetric, Jacobian, Hessian, Wronskian
47. Alternant Matrices and Determinants
48. Elementary and Complete Symmetric Functions
49. Bialternant Symmetric Functions of Jacobi
50. Confluent or Differentiated Alternants
51. Persymmetric, Circulant and Centrosymmetric
52. Dialytic Elimination: Bigradients
53. Continuant Matrices and Continuants
54. Jacobians, Hessians and Wronskians