Table of Contents
Chapter 1 Matrices: Basic Skills and Applications 1
1.1 Definitions, Addition, Scalar Multiplication, and Notation 1
1.2 Parts-Listing and Input-Output Matrices; Triangular, Diagonal, and Symmetric Matrices 8
1.3 Matrix Multiplication and Vector Inner Products 20
1.4 Input-Output Models and Compact Notation 30
1.5 Identities and Inverses 36
*1.6 Using Inverses in Cryptography 44
Chapter 2 Gauss-Jordan Row Operations 49
2.1 Linear Equations with a Unique Solution 49
2.2 Linear Equations with a Unique Solution (continued) 59
2.3 Elementary Matrices 66
2.4 Finding the Multiplicative Inverse of a Matrix 72
2.5 Using Inverses in Leontief Models 78
*2.6 Parts-Listing Problem and Accounting Model 84
Chapter 3 Systems of Linear Equations without Unique Solutions 90
3.1 Recognizing Nonunique Solutions 90
3.2 Finding Nonunique Solutions of a System of Linear Equations 98
*3.3 Analysis of Traffic Flow Networks 109
3.4 Geometric Interpretations of Linear Equations 114
*3.5 Linear Independence and Dependence and Row Rank 123
*Chapter 4 Determinants 131
4.1 Classical Expansion of Determinants 131
4.2 Uses of Determinants 139
4.3 The Gauss-Jordan Method Applied to Determinants 145
Chapter 5 Introduction to Linear Programming 151
5.1 Graphing Linear Inequalities 151
5.2 Setting Up Linear Programming and the Graphical Approach 160
5.3 Tabular Solutions of Linear Programming Problems 173
5.4 Minimum Problems 181
*5.5 The Classic Diet Problem 187
Chapter 6 The Simplex Algorithm 194
6.1 Solving Standard Linear Programming Problems Using the Simplex Algorithm 194
6.2 Why the Simplex Algorithm Works 203
*6.3 Linear Programming Problems That Are Not Standard 212
*6.4 A Model of Cleaning a River at Minimum Cost 225
*Chapter 7 Dual Problems 234
7.1 Definition of Dual Problems and Economic Interpretation 234
7.2 Solving for the Independent Variables in a Dual Problem 240
7.3 Dual Problem Proofs 245
*Chapter 8 The Transportation Problem 250
8.1 Northwest Corner Algorithm and Minimum Cell Algorithm 250
8.2 The Stepping-Stone Algorithm 265
8.3 Harder Stepping Stones 274
8.4 The Assignment Problem 280
Chapter 9 Probability 293
9.1 Basic Concepts 293
*9.2 Counting 306
9.3 Conditional Probability 315
*9.4 Regular Markov Matrices 325
Chapter 10 Game Theory 336
10.1 Expected Value 336
10.2 Saddle Points and Mixed Strategies 342
10.3 Games and Matrices 355
10.4 Solving Matrix Games Using the Simplex Algorithm 364
Appendixes 371
1 Signed Numbers 371
2 Slopes and Graphs of Linear Equations 373
3 Solving Two Simultaneous Equations in Two Unknowns 380
Answers to Sample Tests 383
Glossary 389
Index 391
