Page 10 - Determinants and Their Applications in Mathematical Physics

Contents

Preface v

1 Determinants, First Minors, and Cofactors 1
1.1 Grassmann Exterior Algebra ................. 1
1.2 Determinants .......................... 1
1.3 First Minors and Cofactors .................. 3
1.4 The Product of Two Determinants — 1 ........... 5

2 A Summary of Basic Determinant Theory 7
2.1 Introduction .......................... 7
2.2 Row and Column Vectors ................... 7
2.3 Elementary Formulas ..................... 8
2.3.1 Basic Properties .................... 8
2.3.2 Matrix-Type Products Related to Row and
Column Operations .................. 10
2.3.3 First Minors and Cofactors; Row and Column
Expansions ....................... 12
2.3.4 Alien Cofactors; The Sum Formula ......... 12
2.3.5 Cramer’s Formula ................... 13
2.3.6 The Cofactors of a Zero Determinant ........ 15
2.3.7 The Derivative of a Determinant .......... 15

3 Intermediate Determinant Theory 16
3.1 Cyclic Dislocations and Generalizations ........... 16
3.2 Second and Higher Minors and Cofactors .......... 18