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  • 11T H E D I T I O N

    ElementaryLinearAlgebraApplicationsVersion

    H OW A R D A NT O NProfessor Emeritus, Drexel University

    C H R I S R O R R E SUniversity of Pennsylvania

  • VICE PRESIDENT AND PUBLISHER Laurie RosatoneSENIOR ACQUISITIONS EDITOR David DietzASSOCIATE CONTENT EDITOR Jacqueline SinacoriFREELANCE DEVELOPMENT EDITOR Anne Scanlan-RohrerMARKETING MANAGER Melanie KurkjianEDITORIAL ASSISTANT Michael O’NealSENIOR PRODUCT DESIGNER Thomas KulesaSENIOR PRODUCTION EDITOR Ken SantorSENIOR CONTENT MANAGER Karoline LucianoOPERATIONS MANAGER Melissa EdwardsSENIOR DESIGNER Maddy LesureMEDIA SPECIALIST Laura AbramsPHOTO RESEARCH EDITOR Felicia RuoccoCOPY EDITOR Lilian BradyPRODUCTION SERVICES Carol Sawyer/The Perfect ProofCOVER ART Norm Christiansen

    This book was set in Times New Roman STD by Techsetters, Inc. and printed and bound by QuadGraphics/Versailles. The cover was printed by Quad Graphics/Versailles.

    This book is printed on acid-free paper.

    Copyright 2014, 2010, 2005, 2000, 1994, 1991, 1987, 1984, 1981, 1977, 1973 by Anton Textbooks, Inc.

    All rights reserved. Published simultaneously in Canada.

    No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or byany means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permittedunder Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior writtenpermission of the Publisher, or authorization through payment of the appropriate per-copy fee to theCopyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, website www.copyright.com.Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley &Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, (201) 748-6011, fax (201) 748-6008, websitewww.wiley.com/go/permissions. Best efforts have been made to determine whether the images ofmathematicians shown in the text are in the public domain or properly licensed. If you believe that an errorhas been made, please contact the Permissions Department.

    Evaluation copies are provided to qualified academics and professionals for review purposes only, for use intheir courses during the next academic year. These copies are licensed and may not be sold or transferred toa third party. Upon completion of the review period, please return the evaluation copy to Wiley. Returninstructions and a free of charge return shipping label are available at www.wiley.com/go/returnlabel. Outsideof the United States, please contact your local representative.

    Library of Congress Cataloging-in-Publication Data

    Anton, Howard, author.Elementary linear algebra : applications version / Howard Anton, Chris Rorres. -- 11th edition.

    pages cmIncludes index.ISBN 978-1-118-43441-3 (cloth)

    1. Algebras, Linear--Textbooks. I. Rorres, Chris, author. II. Title.QA184.2.A58 2013512'.5--dc23

    2013033542

    ISBN 978-1-118-43441-3ISBN Binder-Ready Version 978-1-118-47422-8

    Printed in the United States of America

    10 9 8 7 6 5 4 3 2 1

    www.copyright.comhttp://www.wiley.com/go/permissionwww.wiley.com/go/returnlabel

  • A B O U T T H E A U T H O R

    Howard Anton obtained his B.A. from Lehigh University, his M.A. from theUniversity of Illinois, and his Ph.D. from the Polytechnic University of Brooklyn, all inmathematics. In the early 1960s he worked for Burroughs Corporation and AvcoCorporation at Cape Canaveral, Florida, where he was involved with the manned spaceprogram. In 1968 he joined the Mathematics Department at Drexel University, wherehe taught full time until 1983. Since then he has devoted the majority of his time totextbook writing and activities for mathematical associations. Dr. Anton was presidentof the EPADEL Section of the Mathematical Association of America (MAA), served onthe Board of Governors of that organization, and guided the creation of the StudentChapters of the MAA. In addition to various pedagogical articles, he has publishednumerous research papers in functional analysis, approximation theory, and topology.He is best known for his textbooks in mathematics, which are among the most widelyused in the world. There are currently more than 175 versions of his books, includingtranslations into Spanish, Arabic, Portuguese, Italian, Indonesian, French, Japanese,Chinese, Hebrew, and German. For relaxation, Dr. Anton enjoys travel andphotography.

    Chris Rorres earned his B.S. degree from Drexel University and his Ph.D. from theCourant Institute of New York University. He was a faculty member of theDepartment of Mathematics at Drexel University for more than 30 years where, inaddition to teaching, he did applied research in solar engineering, acoustic scattering,population dynamics, computer system reliability, geometry of archaeological sites,optimal animal harvesting policies, and decision theory. He retired from Drexel in 2001as a Professor Emeritus of Mathematics and is now a mathematical consultant. Healso has a research position at the School of Veterinary Medicine at the University ofPennsylvania where he does mathematical modeling of animal epidemics. Dr. Rorres isa recognized expert on the life and work of Archimedes and has appeared in varioustelevision documentaries on that subject. His highly acclaimed website on Archimedes(http://www.math.nyu.edu/~crorres/Archimedes/contents.html) is a virtual book thathas become an important teaching tool in mathematical history for students aroundthe world.

    To:My wife, Pat

    My children, Brian, David, and Lauren

    My parents, Shirley and Benjamin

    My benefactor, Stephen Girard (1750–1831),whose philanthropy changed my life

    Howard Anton

    To:Billie

    Chris Rorres

  • vi

    P R E F A C E

    This textbook is an expanded version ofElementary Linear Algebra, eleventh edition, byHoward Anton. The first nine chapters of this book are identical to the first nine chaptersof that text; the tenth chapter consists of twenty applications of linear algebra drawnfrom business, economics, engineering, physics, computer science, approximation theory,ecology, demography, and genetics. The applications are largely independent of eachother, and each includes a list of mathematical prerequisites. Thus, each instructor hasthe flexibility to choose those applications that are suitable for his or her students and toincorporate each application anywhere in the course after the mathematical prerequisiteshave been satisfied. Chapters 1–9 include simpler treatments of some of the applicationscovered in more depth in Chapter 10.

    This edition gives an introductory treatment of linear algebra that is suitable for afirst undergraduate course. Its aim is to present the fundamentals of linear algebra in theclearest possible way—sound pedagogy is the main consideration. Although calculusis not a prerequisite, there is some optional material that is clearly marked for studentswith a calculus background. If desired, that material can be omitted without loss ofcontinuity.

    Technology is not required to use this text, but for instructors who would like touse MATLAB, Mathematica, Maple, or calculators with linear algebra capabilities, wehave posted some supporting material that can be accessed at either of the followingcompanion websites:

    www.howardanton.comwww.wiley.com/college/anton

    Summary of Changes inThis Edition

    Many parts of the text have been revised based on an extensive set of reviews. Here arethe primary changes:• Earlier Linear Transformations Linear transformations are introduced earlier (starting

    in Section 1.8). Many exercise sets, as well as parts of Chapters 4 and 8, have beenrevised in keeping with the earlier introduction of linear transformations.

    • New Exercises Hundreds of new exercises of all types have been added throughoutthe text.

    • Technology Exercises requiring technology such as MATLAB, Mathematica, or Maplehave been added and supporting data sets have been posted on the companion websitesfor this text. The use of technology is not essential, and these exercises can be omittedwithout affecting the flow of the text.

    • Exercise Sets Reorganized Many multiple-part exercises have been subdivided to createa better balance between odd and even exercise types. To simplify the instructor’s taskof creating assignments, exercise sets have been arranged in clearly defined categories.

    • Reorganization In addition to the earlier introduction of linear transformations, theold Section 4.12 on Dynamical Systems and Markov Chains has been moved to Chap-ter 5 in order to incorporate material on eigenvalues and eigenvectors.

    • Rewriting Section 9.3 on Internet Search Engines from the previous edition has beenrewritten to reflect more accurately how the Google PageRank algorithm works inpractice. That section is now Section 10.20 of the applications version of this text.

    • Appendix A Rewritten The appendix on reading and writing proofs has been expandedand revised to better support courses that focus on proving theorems.

    • Web Materials Supplementary web materials now include various applications mod-ules, three modules on linear programming, and an alternative presentation of deter-minants based on permutations.

    • Applications Chapter Section 10.2 of the previous edition has been moved to thewebsites that accompany this text, so it is now part of a three-module set on Linear

    www.wiley.com/college/anton

  • Preface vii

    Programming. A new section on Internet search engines has been added that explainsthe PageRank algorithm used by Google.

    Hallmark Features • Relationships Among Concepts One of our main pedagogical goals is to convey to thestudent that linear algebra is a cohesive subject and not simply a collection of isolateddefinitions and techniques. One way in which we do this is by using a crescendo ofEquivalent Statements theorems that continually revisit relationships among systemsof equations, matrices, determinants, vectors, linear transformations, and eigenvalues.To get a general sense of how we use this technique see Theorems 1.5.3, 1.6.4, 2.3.8,4.8.8, and then Theorem 5.1.5, for example.

    • Smooth Transition to Abstraction Because the transition from Rn to general vectorspaces is difficult for many students, considerable effort is devoted to explaining thepurpose of abstraction and helping the student to “visualize” abstract ideas by drawinganalogies to familiar geometric ideas.

    • Mathematical Precision When reasonable, we try to be mathematically precise. Inkeeping with the level of student audience, proofs are presented in a patient style thatis tailored for beginners.

    • Suitability for a Diverse Audience This text is designed to serve the needs of studentsin engineering, computer science, biology, physics, business, and economics as well asthose majoring in mathematics.

    • Historical Notes To give the students a sense of mathematical history and to conveythat real people created the mathematical theorems and equations they are studying, wehave included numerous Historical Notes that put the topic being studied in historicalperspective.

    About the Exercises • Graded Exercise Sets Each exercise set in the first nine chapters begins with routinedrill problems and progresses to problems with more substance. These are followedby three categories of exercises, the first focusing on proofs, the second on true/falseexercises, and the third on problems requiring technology. This compartmentalizationis designed to simplify the instructor’s task of selecting exercises for homework.

    • Proof Exercises Linear algebra courses vary widely in their emphasis on proofs, soexercises involving proofs have been grouped and compartmentalized for easy identifi-cation. Appendix A has been rewritten to provide students more guidance on provingtheorems.

    • True/False Exercises The True/False exercises are designed to check conceptual un-derstanding and logical reasoning. To avoid pure guesswork, the students are requiredto justify their responses in some way.

    • Technology Exercises Exercises that require technology have also been grouped. Toavoid burdening the student with keyboarding, the relevant data files have been postedon the websites that accompany this text.

    • Supplementary Exercises Each of the first nine chapters ends with a set of supplemen-tary exercises that draw on all topics in the chapter. These tend to be more challenging.

    Supplementary Materialsfor Students

    • Student Solutions Manual This supplement provides detailed solutions to most odd-numbered exercises (ISBN 978-1-118-464427).

    • Data Files Data files for the technology exercises are posted on the companion websitesthat accompany this text.

    • MATLAB Manual and Linear Algebra Labs This supplement contains a set of MATLABlaboratory projects written by Dan Seth of West Texas A&M University. It is designedto help students learn key linear algebra concepts by using MATLAB and is available inPDF form without charge to students at schools adopting the 11th edition of the text.

    • Videos A complete set of Daniel Solow’sHow toRead andDoProofs videos is availableto students through WileyPLUS as well as the companion websites that accompany

  • viii Preface

    this text. Those materials include a guide to help students locate the lecture videosappropriate for specific proofs in the text.

    Supplementary Materialsfor Instructors

    • Instructor’s Solutions Manual This supplement provides worked-out solutions to mostexercises in the text (ISBN 978-1-118-434482).

    • PowerPoint Presentations PowerPoint slides are provided that display important def-initions, examples, graphics, and theorems in the book. These can also be distributedto students as review materials or to simplify note taking.

    • Test Bank Test questions and sample exams are available in PDF or LATEX form.• WileyPLUS An online environment for effective teaching and learning. WileyPLUS

    builds student confidence by taking the guesswork out of studying and by providing aclear roadmap of what to do, how to do it, and whether it was done right. Its purpose isto motivate and foster initiative so instructors can have a greater impact on classroomachievement and beyond.

    A Guide for the Instructor Although linear algebra courses vary widely in content and philosophy, most coursesfall into two categories—those with about 40 lectures and those with about 30 lectures.Accordingly, we have created long and short templates as possible starting points forconstructing a course outline. Of course, these are just guides, and you will certainlywant to customize them to fit your local interests and requirements. Neither of thesesample templates includes applications or the numerical methods in Chapter 9. Thosecan be added, if desired, and as time permits.

    Long Template Short Template

    Chapter 1: Systems of Linear Equations and Matrices 8 lectures 6 lectures

    Chapter 2: Determinants 3 lectures 2 lectures

    Chapter 3: Euclidean Vector Spaces 4 lectures 3 lectures

    Chapter 4: General Vector Spaces 10 lectures 9 lectures

    Chapter 5: Eigenvalues and Eigenvectors 3 lectures 3 lectures

    Chapter 6: Inner Product Spaces 3 lectures 1 lecture

    Chapter 7: Diagonalization and Quadratic Forms 4 lectures 3 lectures

    Chapter 8: General Linear Transformations 4 lectures 3 lectures

    Total: 39 lectures 30 lectures

    Reviewers The following people reviewed the plans for this edition, critiqued much of the content,and provided me with insightful pedagogical advice:

    John Alongi, Northwestern UniversityJiu Ding, University of Southern MississippiEugene Don, City University of New York at QueensJohn Gilbert, University of Texas AustinDanrun Huang, St. Cloud State UniversityCraig Jensen, University of New OrleansSteve Kahan, City University of New York at QueensHarihar Khanal, Embry-Riddle Aeronautical UniversityFirooz Khosraviyani, Texas A&M International UniversityY. George Lai, Wilfred Laurier UniversityKouok Law, Georgia Perimeter CollegeMark MacLean, Seattle University

  • Preface ix

    Vasileios Maroulas, University of Tennessee, KnoxvilleDaniel Reynolds, Southern Methodist UniversityQin Sheng, Baylor UniversityLaura Smithies, Kent State UniversityLarry Susanka, Bellevue CollegeCristina Tone, University of LouisvilleYvonne Yaz, Milwaukee School of EngineeringRuhan Zhao, State University of New York at Brockport

    Exercise Contributions Special thanks are due to three talented people who worked on various aspects of theexercises:

    Przemyslaw Bogacki, Old Dominion University – who solved the exercises and createdthe solutions manuals.

    Roger Lipsett, Brandeis University – who proofread the manuscript and exercise solu-tions for mathematical accuracy.

    Daniel Solow,CaseWesternReserveUniversity – author of “How toRead andDoProofs,”for providing videos on techniques of proof and a key to using those videos in coordi-nation with this text.

    Sky Pelletier Waterpeace – who critiqued the technology exercises, suggested improve-ments, and provided the data sets.

    Special Contributions I would also like to express my deep appreciation to the following people with whom Iworked on a daily basis:

    Anton Kaul – who worked closely with me at every stage of the project and helped to writesome new text material and exercises. On the many occasions that I needed mathematicalor pedagogical advice, he was the person I turned to. I cannot thank him enough for hisguidance and the many contributions he has made to this edition.

    David Dietz – my editor, for his patience, sound judgment, and dedication to producinga quality book.

    Anne Scanlan-Rohrer – of Two Ravens Editorial, who coordinated the entire project andbrought all of the pieces together.

    Jacqueline Sinacori – who managed many aspects of the content and was always thereto answer my often obscure questions.

    Carol Sawyer – ofThePerfectProof, who managed the myriad of details in the productionprocess and helped with proofreading.

    Maddy Lesure – with whom I have worked for many years and whose elegant sense ofdesign is apparent in the pages of this book.

    Lilian Brady – my copy editor for almost 25 years. I feel fortunate to have been the ben-eficiary of her remarkable knowledge of typography, style, grammar, and mathematics.

    Pat Anton – of Anton Textbooks, Inc., who helped with the mundane chores duplicating,shipping, accuracy checking, and tasks too numerous to mention.

    John Rogosich – of Techsetters, Inc., who programmed the design, managed the compo-sition, and resolved many difficult technical issues.

    Brian Haughwout – of Techsetters, Inc., for his careful and accurate work on the illustra-tions.

    Josh Elkan – for providing valuable assistance in accuracy checking.

    Howard AntonChris Rorres

  • x

    C O N T E N T S

    C H A P T E R 1 Systems of Linear Equations and Matrices 1

    1.1 Introduction to Systems of Linear Equations 21.2 Gaussian Elimination 111.3 Matrices and Matrix Operations 251.4 Inverses; Algebraic Properties of Matrices 391.5 Elementary Matrices and a Method for Finding A−1 521.6 More on Linear Systems and Invertible Matrices 611.7 Diagonal, Triangular, and Symmetric Matrices 671.8 Matrix Transformations 751.9 Applications of Linear Systems 84

    • Network Analysis (Traffic Flow) 84• Electrical Circuits 86• Balancing Chemical Equations 88• Polynomial Interpolation 91

    1.10 Application: Leontief Input-Output Models 96

    C H A P T E R 2 Determinants 105

    2.1 Determinants by Cofactor Expansion 1052.2 Evaluating Determinants by Row Reduction 1132.3 Properties of Determinants; Cramer’s Rule 118

    C H A P T E R 3 EuclideanVector Spaces 131

    3.1 Vectors in 2-Space, 3-Space, and n-Space 1313.2 Norm, Dot Product, and Distance in Rn 1423.3 Orthogonality 1553.4 The Geometry of Linear Systems 1643.5 Cross Product 172

    C H A P T E R 4 General Vector Spaces 183

    4.1 Real Vector Spaces 1834.2 Subspaces 1914.3 Linear Independence 2024.4 Coordinates and Basis 2124.5 Dimension 2214.6 Change of Basis 2294.7 Row Space, Column Space, and Null Space 2374.8 Rank, Nullity, and the Fundamental Matrix Spaces 2484.9 Basic Matrix Transformations in R2 and R3 2594.10 Properties of Matrix Transformations 2704.11 Application: Geometry of Matrix Operators on R2 280

  • Contents xi

    C H A P T E R 5 Eigenvalues and Eigenvectors 291

    5.1 Eigenvalues and Eigenvectors 2915.2 Diagonalization 3025.3 Complex Vector Spaces 3135.4 Application: Differential Equations 3265.5 Application: Dynamical Systems and Markov Chains 332

    C H A P T E R 6 Inner Product Spaces 345

    6.1 Inner Products 3456.2 Angle and Orthogonality in Inner Product Spaces 3556.3 Gram–Schmidt Process; QR-Decomposition 3646.4 Best Approximation; Least Squares 3786.5 Application: Mathematical Modeling Using Least Squares 3876.6 Application: Function Approximation; Fourier Series 394

    C H A P T E R 7 Diagonalization and Quadratic Forms 401

    7.1 Orthogonal Matrices 4017.2 Orthogonal Diagonalization 4097.3 Quadratic Forms 4177.4 Optimization Using Quadratic Forms 4297.5 Hermitian, Unitary, and Normal Matrices 437

    C H A P T E R 8 General Linear Transformations 447

    8.1 General Linear Transformations 4478.2 Compositions and Inverse Transformations 4588.3 Isomorphism 4668.4 Matrices for General Linear Transformations 4728.5 Similarity 481

    C H A P T E R 9 Numerical Methods 491

    9.1 LU-Decompositions 4919.2 The Power Method 5019.3 Comparison of Procedures for Solving Linear Systems 5099.4 Singular Value Decomposition 5149.5 Application: Data Compression Using Singular Value Decomposition 521

    C H A P T E R 10 Applications of Linear Algebra 527

    10.1 Constructing Curves and Surfaces Through Specified Points 52810.2 The Earliest Applications of Linear Algebra 53310.3 Cubic Spline Interpolation 540

  • xii Contents

    10.4 Markov Chains 55110.5 Graph Theory 56110.6 Games of Strategy 57010.7 Leontief Economic Models 57910.8 Forest Management 58810.9 Computer Graphics 59510.10 Equilibrium Temperature Distributions 60310.11 Computed Tomography 61310.12 Fractals 62410.13 Chaos 63910.14 Cryptography 65210.15 Genetics 66310.16 Age-Specific Population Growth 67310.17 Harvesting of Animal Populations 68310.18 A Least Squares Model for Human Hearing 69110.19 Warps and Morphs 69710.20 Internet Search Engines 706

    A P P E N D I X A Working with Proofs A1

    A P P E N D I X B Complex Numbers A5

    Answers to Exercises A13

    Index I1

  • 1

    C H A P T E R 1

    Systems of LinearEquations and Matrices

    CHAPTER CONTENTS 1.1 Introduction to Systems of Linear Equations 2

    1.2 Gaussian Elimination 11

    1.3 Matrices and Matrix Operations 25

    1.4 Inverses; Algebraic Properties of Matrices 39

    1.5 Elementary Matrices and a Method for Finding A−1 521.6 More on Linear Systems and Invertible Matrices 61

    1.7 Diagonal,Triangular, and Symmetric Matrices 67

    1.8 MatrixTransformations 75

    1.9 Applications of Linear Systems 84• Network Analysis (Traffic Flow) 84• Electrical Circuits 86• Balancing Chemical Equations 88• Polynomial Interpolation 91

    1.10 Leontief Input-Output Models 96

    INTRODUCTION Information in science, business, and mathematics is often organized into rows andcolumns to form rectangular arrays called “matrices” (plural of “matrix”). Matricesoften appear as tables of numerical data that arise from physical observations, but theyoccur in various mathematical contexts as well. For example, we will see in this chapterthat all of the information required to solve a system of equations such as

    5x + y = 32x − y = 4

    is embodied in the matrix [5

    2

    1

    −13

    4

    ]and that the solution of the system can be obtained by performing appropriateoperations on this matrix. This is particularly important in developing computerprograms for solving systems of equations because computers are well suited formanipulating arrays of numerical information. However, matrices are not simply anotational tool for solving systems of equations; they can be viewed as mathematicalobjects in their own right, and there is a rich and important theory associated withthem that has a multitude of practical applications. It is the study of matrices andrelated topics that forms the mathematical field that we call “linear algebra.” In thischapter we will begin our study of matrices.

  • 2 Chapter 1 Systems of Linear Equations and Matrices

    1.1 Introduction to Systems of Linear EquationsSystems of linear equations and their solutions constitute one of the major topics that wewill study in this course. In this first section we will introduce some basic terminology anddiscuss a method for solving such systems.

    Linear Equations Recall that in two dimensions a line in a rectangular xy-coordinate system can be repre-sented by an equation of the form

    ax + by = c (a, b not both 0)and in three dimensions a plane in a rectangular xyz-coordinate system can be repre-sented by an equation of the form

    ax + by + cz = d (a, b, c not all 0)These are examples of “linear equations,” the first being a linear equation in the variablesx and y and the second a linear equation in the variables x, y, and z. More generally, wedefine a linear equation in the n variables x1, x2, . . . , xn to be one that can be expressedin the form

    a1x1 + a2x2 + · · · + anxn = b (1)where a1, a2, . . . , an and b are constants, and the a’s are not all zero. In the special caseswhere n = 2 or n = 3, we will often use variables without subscripts and write linearequations as

    a1x + a2y = b (a1, a2 not both 0) (2)a1x + a2y + a3z = b (a1, a2, a3 not all 0) (3)

    In the special case where b = 0, Equation (1) has the forma1x1 + a2x2 + · · · + anxn = 0 (4)

    which is called a homogeneous linear equation in the variables x1, x2, . . . , xn.

    EXAMPLE 1 Linear Equations

    Observe that a linear equation does not involve any products or roots of variables. Allvariables occur only to the first power and do not appear, for example, as arguments oftrigonometric, logarithmic, or exponential functions. The following are linear equations:

    x + 3y = 7 x1 − 2x2 − 3x3 + x4 = 012 x − y + 3z = −1 x1 + x2 + · · · + xn = 1

    The following are not linear equations:

    x + 3y2 = 4 3x + 2y − xy = 5sin x + y = 0 √x1 + 2x2 + x3 = 1

    A finite set of linear equations is called a system of linear equations or, more briefly,a linear system. The variables are called unknowns. For example, system (5) that followshas unknowns x and y, and system (6) has unknowns x1, x2, and x3.

    5x + y = 3 4x1 − x2 + 3x3 = −12x − y = 4 3x1 + x2 + 9x3 = −4 (5–6)

  • 1.1 Introduction to Systems of Linear Equations 3

    A general linear system of m equations in the n unknowns x1, x2, . . . , xn can be writtenThe double subscripting onthe coefficients aij of the un-knowns gives their locationin the system—the first sub-script indicates the equationin which the coefficient occurs,and the second indicates whichunknown it multiplies. Thus,a12 is in the first equation andmultiplies x2.

    asa11x1 + a12x2 + · · · + a1nxn = b1a21x1 + a22x2 + · · · + a2nxn = b2

    ......

    ......

    am1x1 + am2x2 + · · · + amnxn = bm(7)

    A solution of a linear system in n unknowns x1, x2, . . . , xn is a sequence of n numberss1, s2, . . . , sn for which the substitution

    x1 = s1, x2 = s2, . . . , xn = snmakes each equation a true statement. For example, the system in (5) has the solution

    x = 1, y = −2and the system in (6) has the solution

    x1 = 1, x2 = 2, x3 = −1These solutions can be written more succinctly as

    (1,−2) and (1, 2,−1)in which the names of the variables are omitted. This notation allows us to interpretthese solutions geometrically as points in two-dimensional and three-dimensional space.More generally, a solution

    x1 = s1, x2 = s2, . . . , xn = snof a linear system in n unknowns can be written as

    (s1, s2, . . . , sn)

    which is called an ordered n-tuple. With this notation it is understood that all variablesappear in the same order in each equation. If n = 2, then the n-tuple is called an orderedpair, and if n = 3, then it is called an ordered triple.

    Linear Systems inTwo andThree Unknowns

    Linear systems in two unknowns arise in connection with intersections of lines. Forexample, consider the linear system

    a1x + b1y = c1a2x + b2y = c2

    in which the graphs of the equations are lines in the xy-plane. Each solution (x, y) of thissystem corresponds to a point of intersection of the lines, so there are three possibilities(Figure 1.1.1):

    1. The lines may be parallel and distinct, in which case there is no intersection andconsequently no solution.

    2. The lines may intersect at only one point, in which case the system has exactly onesolution.

    3. The lines may coincide, in which case there are infinitely many points of intersection(the points on the common line) and consequently infinitely many solutions.

    In general, we say that a linear system is consistent if it has at least one solution andinconsistent if it has no solutions. Thus, a consistent linear systemof two equations in

  • 4 Chapter 1 Systems of Linear Equations and Matrices

    Figure 1.1.1

    x

    y

    No solution

    x

    y

    One solution

    x

    y

    Infinitely manysolutions

    (coincident lines)

    two unknowns has either one solution or infinitely many solutions—there are no otherpossibilities. The same is true for a linear system of three equations in three unknowns

    a1x + b1y + c1z = d1a2x + b2y + c2z = d2a3x + b3y + c3z = d3

    in which the graphs of the equations are planes. The solutions of the system, if any,correspond to points where all three planes intersect, so again we see that there are onlythree possibilities—no solutions, one solution, or infinitely many solutions (Figure 1.1.2).

    No solutions(three parallel planes;

    no common intersection)

    No solutions(two parallel planes;

    no common intersection)

    No solutions(no common intersection)

    Infinitely many solutions(planes are all coincident;

    intersection is a plane)

    Infinitely many solutions(intersection is a line)

    One solution(intersection is a point)

    No solutions(two coincident planesparallel to the third;

    no common intersection)

    Infinitely many solutions(two coincident planes;intersection is a line)

    Figure 1.1.2

    We will prove later that our observations about the number of solutions of linearsystems of two equations in two unknowns and linear systems of three equations inthree unknowns actually hold for all linear systems. That is:

    Every system of linear equations has zero, one, or infinitely many solutions. There areno other possibilities.

  • 1.1 Introduction to Systems of Linear Equations 5

    EXAMPLE 2 A Linear System with One Solution

    Solve the linear systemx − y = 1

    2x + y = 6Solution We can eliminate x from the second equation by adding −2 times the firstequation to the second. This yields the simplified system

    x − y = 13y = 4

    From the second equation we obtain y = 43 , and on substituting this value in the firstequation we obtain x = 1 + y = 73 . Thus, the system has the unique solution

    x = 73 , y = 43Geometrically, this means that the lines represented by the equations in the systemintersect at the single point

    (73 ,

    43

    ). We leave it for you to check this by graphing the

    lines.

    EXAMPLE 3 A Linear System with No Solutions

    Solve the linear systemx + y = 4

    3x + 3y = 6Solution We can eliminate x from the second equation by adding −3 times the firstequation to the second equation. This yields the simplified system

    x + y = 40 = −6

    The second equation is contradictory, so the given system has no solution. Geometrically,this means that the lines corresponding to the equations in the original system are paralleland distinct. We leave it for you to check this by graphing the lines or by showing thatthey have the same slope but different y-intercepts.

    EXAMPLE 4 A Linear System with Infinitely Many Solutions

    Solve the linear system4x − 2y = 1

    16x − 8y = 4Solution We can eliminate x from the second equation by adding −4 times the firstequation to the second. This yields the simplified system

    4x − 2y = 10 = 0

    The second equation does not impose any restrictions on x and y and hence can beomitted. Thus, the solutions of the system are those values of x and y that satisfy thesingle equation

    4x − 2y = 1 (8)Geometrically, this means the lines corresponding to the two equations in the originalsystem coincide. One way to describe the solution set is to solve this equation for x interms of y to obtain x = 14 + 12 y and then assign an arbitrary value t (called a parameter)

  • 6 Chapter 1 Systems of Linear Equations and Matrices

    to y. This allows us to express the solution by the pair of equations (called parametricequations)

    x = 14 + 12 t, y = tWe can obtain specific numerical solutions from these equations by substituting numer-

    In Example 4 we could havealso obtained parametricequations for the solutionsby solving (8) for y in termsof x and letting x = t bethe parameter. The resultingparametric equations wouldlook different but woulddefine the same solution set.

    ical values for the parameter t . For example, t = 0 yields the solution ( 14 , 0) , t = 1yields the solution

    (34 , 1

    ), and t = −1 yields the solution (− 14 ,−1) . You can confirm

    that these are solutions by substituting their coordinates into the given equations.

    EXAMPLE 5 A Linear System with Infinitely Many Solutions

    Solve the linear systemx − y + 2z = 5

    2x − 2y + 4z = 103x − 3y + 6z = 15

    Solution This system can be solved by inspection, since the second and third equationsare multiples of the first. Geometrically, this means that the three planes coincide andthat those values of x, y, and z that satisfy the equation

    x − y + 2z = 5 (9)automatically satisfy all three equations. Thus, it suffices to find the solutions of (9).We can do this by first solving this equation for x in terms of y and z, then assigningarbitrary values r and s (parameters) to these two variables, and then expressing thesolution by the three parametric equations

    x = 5 + r − 2s, y = r, z = sSpecific solutions can be obtained by choosing numerical values for the parameters rand s. For example, taking r = 1 and s = 0 yields the solution (6, 1, 0).

    Augmented Matrices andElementary Row Operations

    As the number of equations and unknowns in a linear system increases, so does thecomplexity of the algebra involved in finding solutions. The required computations canbe made more manageable by simplifying notation and standardizing procedures. Forexample, by mentally keeping track of the location of the +’s, the x’s, and the =’s in thelinear system

    a11x1 + a12x2 + · · ·+ a1nxn = b1a21x1 + a22x2 + · · ·+ a2nxn = b2

    ......

    ......

    am1x1 + am2x2 + · · ·+ amnxn = bmwe can abbreviate the system by writing only the rectangular array of numbers⎡

    ⎢⎢⎢⎣a11 a12 · · · a1n b1a21 a22 · · · a2n b2...

    ......

    ...am1 am2 · · · amn bm

    ⎤⎥⎥⎥⎦

    This is called the augmented matrix for the system. For example, the augmented matrix

    As noted in the introductionto this chapter, the term “ma-trix” is used in mathematics todenote a rectangular array ofnumbers. In a later sectionwe will study matrices in de-tail, but for now we will onlybe concerned with augmentedmatrices for linear systems.

    for the system of equations

    x1 + x2 + 2x3 = 92x1 + 4x2 − 3x3 = 13x1 + 6x2 − 5x3 = 0

    is

    ⎡⎢⎣1 1 2 92 4 −3 1

    3 6 −5 0

    ⎤⎥⎦

  • 1.1 Introduction to Systems of Linear Equations 7

    The basic method for solving a linear system is to perform algebraic operations onthe system that do not alter the solution set and that produce a succession of increasinglysimpler systems, until a point is reached where it can be ascertained whether the systemis consistent, and if so, what its solutions are. Typically, the algebraic operations are:

    1. Multiply an equation through by a nonzero constant.

    2. Interchange two equations.

    3. Add a constant times one equation to another.

    Since the rows (horizontal lines) of an augmented matrix correspond to the equations inthe associated system, these three operations correspond to the following operations onthe rows of the augmented matrix:

    1. Multiply a row through by a nonzero constant.

    2. Interchange two rows.

    3. Add a constant times one row to another.

    These are called elementary row operations on a matrix.In the following example we will illustrate how to use elementary row operations and

    an augmented matrix to solve a linear system in three unknowns. Since a systematicprocedure for solving linear systems will be developed in the next section, do not worryabout how the steps in the example were chosen. Your objective here should be simplyto understand the computations.

    EXAMPLE 6 Using Elementary Row Operations

    In the left column we solve a system of linear equations by operating on the equations inthe system, and in the right column we solve the same system by operating on the rowsof the augmented matrix.

    x + y + 2z = 92x + 4y − 3z = 13x + 6y − 5z = 0

    ⎡⎢⎣1 1 2 92 4 −3 1

    3 6 −5 0

    ⎤⎥⎦

    Add −2 times the first equation to the secondto obtain

    x + y + 2z = 92y − 7z = −17

    3x + 6y − 5z = 0

    Add −2 times the first row to the second toobtain ⎡

    ⎢⎣1 1 2 90 2 −7 −173 6 −5 0

    ⎤⎥⎦

    Maxime Bôcher(1867–1918)

    Historical Note The first known use of augmented matrices appearedbetween 200 B.C. and 100 B.C. in a Chinese manuscript entitled NineChapters of Mathematical Art. The coefficients were arranged incolumns rather than in rows, as today, but remarkably the system wassolved by performing a succession of operations on the columns. Theactual use of the term augmented matrix appears to have been intro-duced by the American mathematician Maxime Bôcher in his book In-troduction to HigherAlgebra, published in 1907. In addition to being anoutstanding research mathematician and an expert in Latin, chemistry,philosophy, zoology, geography, meteorology, art, and music, Bôcherwas an outstanding expositor of mathematics whose elementary text-books were greatly appreciated by students and are still in demandtoday.

    [Image: Courtesy of the American Mathematical Societywww.ams.org]

  • 8 Chapter 1 Systems of Linear Equations and Matrices

    Add −3 times the first equation to the third toobtain

    x + y + 2z = 92y − 7z = −173y − 11z = −27

    Add−3 times the first row to the third to obtain⎡⎢⎣1 1 2 90 2 −7 −17

    0 3 −11 −27

    ⎤⎥⎦

    Multiply the second equation by 12 to obtain

    x + y + 2z = 9y − 72z = − 172

    3y − 11z = −27

    Multiply the second row by 12 to obtain⎡⎢⎣1 1 2 90 1 − 72 − 172

    0 3 −11 −27

    ⎤⎥⎦

    Add −3 times the second equation to the thirdto obtain

    x + y + 2z = 9y − 72z = − 172− 12z = − 32

    Add −3 times the second row to the third toobtain ⎡

    ⎢⎢⎣1 1 2 9

    0 1 − 72 − 1720 0 − 12 − 32

    ⎤⎥⎥⎦

    Multiply the third equation by −2 to obtainx + y + 2z = 9

    y − 72z = − 172z = 3

    Multiply the third row by −2 to obtain⎡⎢⎣1 1 2 90 1 − 72 − 172

    0 0 1 3

    ⎤⎥⎦

    Add −1 times the second equation to the firstto obtain

    x + 112 z = 352y − 72z = − 172

    z = 3

    Add −1 times the second row to the first toobtain ⎡

    ⎢⎢⎣1 0 112

    352

    0 1 − 72 − 1720 0 1 3

    ⎤⎥⎥⎦

    Add −112 times the third equation to the firstand 72 times the third equation to the second toobtain

    x = 1y = 2

    z = 3

    Add − 112 times the third row to the first and 72times the third row to the second to obtain⎡

    ⎢⎣1 0 0 10 1 0 20 0 1 3

    ⎤⎥⎦

    The solution x = 1, y = 2, z = 3 is now evident.

    The solution in this examplecan also be expressed as the or-dered triple (1, 2, 3) with theunderstanding that the num-bers in the triple are in thesame order as the variables inthe system, namely, x, y, z.

    Exercise Set 1.11. In each part, determine whether the equation is linear in x1,

    x2, and x3.

    (a) x1 + 5x2 −√

    2 x3 = 1 (b) x1 + 3x2 + x1x3 = 2(c) x1 = −7x2 + 3x3 (d) x−21 + x2 + 8x3 = 5(e) x3/51 − 2x2 + x3 = 4 (f ) πx1 −

    √2 x2 = 71/3

    2. In each part, determine whether the equation is linear in xand y.

    (a) 21/3x +√3y = 1 (b) 2x1/3 + 3√y = 1(c) cos

    7

    )x − 4y = log 3 (d) π7 cos x − 4y = 0

    (e) xy = 1 (f ) y + 7 = x

  • 1.1 Introduction to Systems of Linear Equations 9

    3. Using the notation of Formula (7), write down a general linearsystem of

    (a) two equations in two unknowns.

    (b) three equations in three unknowns.

    (c) two equations in four unknowns.

    4. Write down the augmented matrix for each of the linear sys-tems in Exercise 3.

    In each part of Exercises 5–6, find a linear system in the un-knowns x1, x2, x3, . . . , that corresponds to the given augmentedmatrix.

    5. (a)

    ⎡⎢⎣2 0 03 −4 0

    0 1 1

    ⎤⎥⎦ (b)

    ⎡⎢⎣3 0 −2 57 1 4 −3

    0 −2 1 7

    ⎤⎥⎦

    6. (a)

    [0 3 −1 −1 −15 2 0 −3 −6

    ]

    (b)

    ⎡⎢⎢⎢⎣

    3 0 1 −4 3−4 0 4 1 −3−1 3 0 −2 −9

    0 0 0 −1 −2

    ⎤⎥⎥⎥⎦

    In each part of Exercises 7–8, find the augmented matrix forthe linear system.

    7. (a) −2x1 = 63x1 = 89x1 = −3

    (b) 6x1 − x2 + 3x3 = 45x2 − x3 = 1

    (c) 2x2 − 3x4 + x5 = 0−3x1 − x2 + x3 = −1

    6x1 + 2x2 − x3 + 2x4 − 3x5 = 6

    8. (a) 3x1 − 2x2 = −14x1 + 5x2 = 37x1 + 3x2 = 2

    (b) 2x1 + 2x3 = 13x1 − x2 + 4x3 = 76x1 + x2 − x3 = 0

    (c) x1 = 1x2 = 2

    x3 = 39. In each part, determine whether the given 3-tuple is a solution

    of the linear system

    2x1 − 4x2 − x3 = 1x1 − 3x2 + x3 = 1

    3x1 − 5x2 − 3x3 = 1(a) (3, 1, 1) (b) (3,−1, 1) (c) (13, 5, 2)(d)

    (132 ,

    52 , 2

    )(e) (17, 7, 5)

    10. In each part, determine whether the given 3-tuple is a solutionof the linear system

    x + 2y − 2z = 33x − y + z = 1−x + 5y − 5z = 5

    (a)(

    57 ,

    87 , 1

    )(b)

    (57 ,

    87 , 0

    )(c) (5, 8, 1)

    (d)(

    57 ,

    107 ,

    27

    )(e)

    (57 ,

    227 , 2

    )11. In each part, solve the linear system, if possible, and use the

    result to determine whether the lines represented by the equa-tions in the system have zero, one, or infinitely many points ofintersection. If there is a single point of intersection, give itscoordinates, and if there are infinitely many, find parametricequations for them.

    (a) 3x − 2y = 46x − 4y = 9

    (b) 2x − 4y = 14x − 8y = 2

    (c) x − 2y = 0x − 4y = 8

    12. Under what conditions on a and b will the following linearsystem have no solutions, one solution, infinitely many solu-tions?

    2x − 3y = a4x − 6y = b

    In each part of Exercises 13–14, use parametric equations todescribe the solution set of the linear equation.

    13. (a) 7x − 5y = 3(b) 3x1 − 5x2 + 4x3 = 7(c) −8x1 + 2x2 − 5x3 + 6x4 = 1(d) 3v − 8w + 2x − y + 4z = 0

    14. (a) x + 10y = 2(b) x1 + 3x2 − 12x3 = 3(c) 4x1 + 2x2 + 3x3 + x4 = 20(d) v + w + x − 5y + 7z = 0In Exercises 15–16, each linear system has infinitely many so-

    lutions. Use parametric equations to describe its solution set.

    15. (a) 2x − 3y = 16x − 9y = 3

    (b) x1 + 3x2 − x3 = −43x1 + 9x2 − 3x3 = −12−x1 − 3x2 + x3 = 4

    16. (a) 6x1 + 2x2 = −83x1 + x2 = −4

    (b) 2x − y + 2z = −46x − 3y + 6z = −12

    −4x + 2y − 4z = 8In Exercises 17–18, find a single elementary row operation that

    will create a 1 in the upper left corner of the given augmented ma-trix and will not create any fractions in its first row.

    17. (a)

    ⎡⎣−3 −1 2 42 −3 3 2

    0 2 −3 1

    ⎤⎦ (b)

    ⎡⎣0 −1 −5 02 −9 3 2

    1 4 −3 3

    ⎤⎦

    18. (a)

    ⎡⎣ 2 4 −6 87 1 4 3−5 4 2 7

    ⎤⎦ (b)

    ⎡⎣ 7 −4 −2 23 −1 8 1−6 3 −1 4

    ⎤⎦

  • 10 Chapter 1 Systems of Linear Equations and Matrices

    In Exercises 19–20, find all values of k for which the givenaugmented matrix corresponds to a consistent linear system.

    19. (a)[

    1 k −44 8 2

    ](b)

    [1 k −14 8 −4

    ]

    20. (a)[

    3 −4 k−6 8 5

    ](b)

    [k 1 −24 −1 2

    ]21. The curve y = ax2 + bx + c shown in the accompanying fig-

    ure passes through the points (x1, y1), (x2, y2), and (x3, y3).Show that the coefficients a, b, and c form a solution of thesystem of linear equations whose augmented matrix is⎡

    ⎢⎣x21 x1 1 y1

    x22 x2 1 y2

    x23 x3 1 y3

    ⎤⎥⎦

    y

    x

    y = ax2 + bx + c

    (x1, y1)

    (x3, y3)

    (x2, y2)

    Figure Ex-21

    22. Explain why each of the three elementary row operations doesnot affect the solution set of a linear system.

    23. Show that if the linear equations

    x1 + kx2 = c and x1 + lx2 = dhave the same solution set, then the two equations are identical(i.e., k = l and c = d).

    24. Consider the system of equations

    ax + by = kcx + dy = lex + fy = m

    Discuss the relative positions of the lines ax + by = k,cx + dy = l, and ex + fy = m when(a) the system has no solutions.

    (b) the system has exactly one solution.

    (c) the system has infinitely many solutions.

    25. Suppose that a certain diet calls for 7 units of fat, 9 units ofprotein, and 16 units of carbohydrates for the main meal, andsuppose that an individual has three possible foods to choosefrom to meet these requirements:

    Food 1: Each ounce contains 2 units of fat, 2 units ofprotein, and 4 units of carbohydrates.

    Food 2: Each ounce contains 3 units of fat, 1 unit ofprotein, and 2 units of carbohydrates.

    Food 3: Each ounce contains 1 unit of fat, 3 units ofprotein, and 5 units of carbohydrates.

    Let x, y, and z denote the number of ounces of the first, sec-ond, and third foods that the dieter will consume at the mainmeal. Find (but do not solve) a linear system in x, y, and zwhose solution tells how many ounces of each food must beconsumed to meet the diet requirements.

    26. Suppose that you want to find values for a, b, and c such thatthe parabola y = ax2 + bx + c passes through the points(1, 1), (2, 4), and (−1, 1). Find (but do not solve) a systemof linear equations whose solutions provide values for a, b,and c. How many solutions would you expect this system ofequations to have, and why?

    27. Suppose you are asked to find three real numbers such that thesum of the numbers is 12, the sum of two times the first plusthe second plus two times the third is 5, and the third numberis one more than the first. Find (but do not solve) a linearsystem whose equations describe the three conditions.

    True-False Exercises

    TF. In parts (a)–(h) determine whether the statement is true orfalse, and justify your answer.

    (a) A linear system whose equations are all homogeneous mustbe consistent.

    (b) Multiplying a row of an augmented matrix through by zero isan acceptable elementary row operation.

    (c) The linear systemx − y = 3

    2x − 2y = kcannot have a unique solution, regardless of the value of k.

    (d) A single linear equation with two or more unknowns musthave infinitely many solutions.

    (e) If the number of equations in a linear system exceeds the num-ber of unknowns, then the system must be inconsistent.

    (f ) If each equation in a consistent linear system is multipliedthrough by a constant c, then all solutions to the new systemcan be obtained by multiplying solutions from the originalsystem by c.

    (g) Elementary row operations permit one row of an augmentedmatrix to be subtracted from another.

    (h) The linear system with corresponding augmented matrix[2 −1 40 0 −1

    ]is consistent.

    Working withTechnology

    T1. Solve the linear systems in Examples 2, 3, and 4 to see howyour technology utility handles the three types of systems.

    T2. Use the result in Exercise 21 to find values of a, b, and cfor which the curve y = ax2 + bx + c passes through the points(−1, 1, 4), (0, 0, 8), and (1, 1, 7).

  • 1.2 Gaussian Elimination 11

    1.2 Gaussian EliminationIn this section we will develop a systematic procedure for solving systems of linearequations. The procedure is based on the idea of performing certain operations on the rowsof the augmented matrix that simplify it to a form from which the solution of the systemcan be ascertained by inspection.

    Considerations in SolvingLinear Systems

    When considering methods for solving systems of linear equations, it is important todistinguish between large systems that must be solved by computer and small systemsthat can be solved by hand. For example, there are many applications that lead tolinear systems in thousands or even millions of unknowns. Large systems require specialtechniques to deal with issues of memory size, roundoff errors, solution time, and soforth. Such techniques are studied in the field of numerical analysis and will only betouched on in this text. However, almost all of the methods that are used for largesystems are based on the ideas that we will develop in this section.

    Echelon Forms In Example 6 of the last section, we solved a linear system in the unknowns x, y, and zby reducing the augmented matrix to the form⎡

    ⎢⎣1 0 0 10 1 0 20 0 1 3

    ⎤⎥⎦

    from which the solution x = 1, y = 2, z = 3 became evident. This is an example of amatrix that is in reduced row echelon form. To be of this form, a matrix must have thefollowing properties:

    1. If a row does not consist entirely of zeros, then the first nonzero number in the rowis a 1. We call this a leading 1.

    2. If there are any rows that consist entirely of zeros, then they are grouped together atthe bottom of the matrix.

    3. In any two successive rows that do not consist entirely of zeros, the leading 1 in thelower row occurs farther to the right than the leading 1 in the higher row.

    4. Each column that contains a leading 1 has zeros everywhere else in that column.

    A matrix that has the first three properties is said to be in row echelon form. (Thus,a matrix in reduced row echelon form is of necessity in row echelon form, but notconversely.)

    EXAMPLE 1 Row Echelon and Reduced Row Echelon Form

    The following matrices are in reduced row echelon form.⎡⎢⎣1 0 0 40 1 0 7

    0 0 1 −1

    ⎤⎥⎦ ,

    ⎡⎢⎣1 0 00 1 0

    0 0 1

    ⎤⎥⎦ ,

    ⎡⎢⎢⎢⎣

    0 1 −2 0 10 0 0 1 3

    0 0 0 0 0

    0 0 0 0 0

    ⎤⎥⎥⎥⎦ ,

    [0 0

    0 0

    ]

    The following matrices are in row echelon form but not reduced row echelon form.⎡⎢⎣1 4 −3 70 1 6 2

    0 0 1 5

    ⎤⎥⎦ ,

    ⎡⎢⎣1 1 00 1 0

    0 0 0

    ⎤⎥⎦ ,

    ⎡⎢⎣0 1 2 6 00 0 1 −1 0

    0 0 0 0 1

    ⎤⎥⎦

  • 12 Chapter 1 Systems of Linear Equations and Matrices

    EXAMPLE 2 More on Row Echelon and Reduced Row Echelon Form

    As Example 1 illustrates, a matrix in row echelon form has zeros below each leading 1,whereas a matrix in reduced row echelon form has zeros below and above each leading1. Thus, with any real numbers substituted for the ∗’s, all matrices of the following typesare in row echelon form:⎡⎢⎢⎢⎣

    1 ∗ ∗ ∗0 1 ∗ ∗0 0 1 ∗0 0 0 1

    ⎤⎥⎥⎥⎦ ,

    ⎡⎢⎢⎢⎣

    1 ∗ ∗ ∗0 1 ∗ ∗0 0 1 ∗0 0 0 0

    ⎤⎥⎥⎥⎦ ,

    ⎡⎢⎢⎢⎣

    1 ∗ ∗ ∗0 1 ∗ ∗0 0 0 0

    0 0 0 0

    ⎤⎥⎥⎥⎦ ,

    ⎡⎢⎢⎢⎢⎢⎣

    0 1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗0 0 0 1 ∗ ∗ ∗ ∗ ∗ ∗0 0 0 0 1 ∗ ∗ ∗ ∗ ∗0 0 0 0 0 1 ∗ ∗ ∗ ∗0 0 0 0 0 0 0 0 1 ∗

    ⎤⎥⎥⎥⎥⎥⎦

    All matrices of the following types are in reduced row echelon form:

    ⎡⎢⎢⎢⎣

    1 0 0 0

    0 1 0 0

    0 0 1 0

    0 0 0 1

    ⎤⎥⎥⎥⎦ ,

    ⎡⎢⎢⎢⎣

    1 0 0 ∗0 1 0 ∗0 0 1 ∗0 0 0 0

    ⎤⎥⎥⎥⎦ ,

    ⎡⎢⎢⎢⎣

    1 0 ∗ ∗0 1 ∗ ∗0 0 0 0

    0 0 0 0

    ⎤⎥⎥⎥⎦ ,

    ⎡⎢⎢⎢⎢⎢⎣

    0 1 ∗ 0 0 0 ∗ ∗ 0 ∗0 0 0 1 0 0 ∗ ∗ 0 ∗0 0 0 0 1 0 ∗ ∗ 0 ∗0 0 0 0 0 1 ∗ ∗ 0 ∗0 0 0 0 0 0 0 0 1 ∗

    ⎤⎥⎥⎥⎥⎥⎦

    If, by a sequence of elementary row operations, the augmented matrix for a system oflinear equations is put in reduced row echelon form, then the solution set can be obtainedeither by inspection or by converting certain linear equations to parametric form. Hereare some examples.

    EXAMPLE 3 Unique Solution

    Suppose that the augmented matrix for a linear system in the unknowns x1, x2, x3, andx4 has been reduced by elementary row operations to⎡

    ⎢⎢⎢⎣1 0 0 0 3

    0 1 0 0 −10 0 1 0 0

    0 0 0 1 5

    ⎤⎥⎥⎥⎦

    This matrix is in reduced row echelon form and corresponds to the equations

    x1 = 3x2 = −1

    x3 = 0x4 = 5

    Thus, the system has a unique solution, namely, x1 = 3, x2 = −1, x3 = 0, x4 = 5.

    In Example 3 we could, ifdesired, express the solutionmore succinctly as the 4-tuple(3,−1, 0, 5).

    EXAMPLE 4 Linear Systems inThree Unknowns

    In each part, suppose that the augmented matrix for a linear system in the unknownsx, y, and z has been reduced by elementary row operations to the given reduced rowechelon form. Solve the system.

    (a)

    ⎡⎢⎣1 0 0 00 1 2 0

    0 0 0 1

    ⎤⎥⎦ (b)

    ⎡⎢⎣1 0 3 −10 1 −4 2

    0 0 0 0

    ⎤⎥⎦ (c)

    ⎡⎢⎣1 −5 1 40 0 0 0

    0 0 0 0

    ⎤⎥⎦

  • 1.2 Gaussian Elimination 13

    Solution (a) The equation that corresponds to the last row of the augmented matrix is

    0x + 0y + 0z = 1Since this equation is not satisfied by any values of x, y, and z, the system is inconsistent.

    Solution (b) The equation that corresponds to the last row of the augmented matrix is

    0x + 0y + 0z = 0This equation can be omitted since it imposes no restrictions on x, y, and z; hence, thelinear system corresponding to the augmented matrix is

    x + 3z = −1y − 4z = 2

    Since x and y correspond to the leading 1’s in the augmented matrix, we call thesethe leading variables. The remaining variables (in this case z) are called free variables.Solving for the leading variables in terms of the free variables gives

    x = −1 − 3zy = 2 + 4z

    From these equations we see that the free variable z can be treated as a parameter andassigned an arbitrary value t , which then determines values for x and y. Thus, thesolution set can be represented by the parametric equations

    x = −1 − 3t, y = 2 + 4t, z = tBy substituting various values for t in these equations we can obtain various solutionsof the system. For example, setting t = 0 yields the solution

    x = −1, y = 2, z = 0and setting t = 1 yields the solution

    x = −4, y = 6, z = 1Solution (c) As explained in part (b), we can omit the equations corresponding to thezero rows, in which case the linear system associated with the augmented matrix consistsof the single equation

    x − 5y + z = 4 (1)from which we see that the solution set is a plane in three-dimensional space. Although(1) is a valid form of the solution set, there are many applications in which it is preferableto express the solution set in parametric form. We can convert (1) to parametric form

    We will usually denote pa-rameters in a general solutionby the letters r, s, t, . . . , butany letters that do not con-flict with the names of theunknowns can be used. Forsystems with more than threeunknowns, subscripted letterssuch as t1, t2, t3, . . . are conve-nient.

    by solving for the leading variable x in terms of the free variables y and z to obtain

    x = 4 + 5y − zFrom this equation we see that the free variables can be assigned arbitrary values, sayy = s and z = t , which then determine the value of x. Thus, the solution set can beexpressed parametrically as

    x = 4 + 5s − t, y = s, z = t (2)Formulas, such as (2), that express the solution set of a linear system parametrically

    have some associated terminology.

    DEFINITION1 If a linear system has infinitely many solutions, then a set of parametricequations from which all solutions can be obtained by assigning numerical values tothe parameters is called a general solution of the system.

  • 14 Chapter 1 Systems of Linear Equations and Matrices

    Elimination Methods We have just seen how easy it is to solve a system of linear equations once its augmentedmatrix is in reduced row echelon form. Now we will give a step-by-step eliminationprocedure that can be used to reduce any matrix to reduced row echelon form. As westate each step in the procedure, we illustrate the idea by reducing the following matrixto reduced row echelon form.⎡

    ⎢⎣0 0 −2 0 7 122 4 −10 6 12 282 4 −5 6 −5 −1

    ⎤⎥⎦

    Step 1. Locate the leftmost column that does not consist entirely of zeros.

    ⎡⎢⎣

    0 0 2 0 7 122 4 10 6 12 282 4 5 6 5 1

    ⎤⎥⎦

    Leftmost nonzero column

    Step 2. Interchange the top row with another row, if necessary, to bring a nonzero entryto the top of the column found in Step 1.⎡

    ⎢⎣2 4 −10 6 12 280 0 −2 0 7 122 4 −5 6 −5 −1

    ⎤⎥⎦ The first and second rows in the preceding

    matrix were interchanged.

    Step 3. If the entry that is now at the top of the column found in Step 1 is a, multiplythe first row by 1/a in order to introduce a leading 1.⎡

    ⎢⎣1 2 −5 3 6 140 0 −2 0 7 122 4 −5 6 −5 −1

    ⎤⎥⎦ The first row of the preceding matrix was

    multiplied by 12 .

    Step 4. Add suitable multiples of the top row to the rows below so that all entries belowthe leading 1 become zeros.⎡

    ⎢⎣1 2 −5 3 6 140 0 −2 0 7 120 0 5 0 −17 −29

    ⎤⎥⎦ −2 times the first row of the preceding

    matrix was added to the third row.

    Step 5. Now cover the top row in the matrix and begin again with Step 1 applied to thesubmatrix that remains. Continue in this way until the entire matrix is in rowechelon form.

    ⎡⎢⎣

    1 2 5 3 6 14

    0 0 1 0 72

    6

    0 0 5 0 17 29

    ⎤⎥⎦ The first row in the submatrix was

    multiplied by 12

    to introduce aleading 1.

    ⎡⎢⎣

    1 2 5 3 6 14

    0 0 2 0 7 12

    0 0 5 0 17 29

    ⎤⎥⎦

    Leftmost nonzero columnin the submatrix

  • 1.2 Gaussian Elimination 15

    ⎡⎢⎣

    1 2 5 3 6 14

    0 0 1 0 72

    6

    0 0 0 0 12

    1

    ⎤⎥⎦ The top row in the submatrix was

    covered, and we returned again toStep 1.

    Leftmost nonzero columnin the new submatrix

    ⎡⎢⎣

    1 2 5 3 6 14

    0 0 1 0 72

    6

    0 0 0 0 12

    1

    ⎤⎥⎦ –5 times the first row of the submatrix

    was added to the second row of thesubmatrix to introduce a zero belowthe leading 1.

    ⎡⎢⎣

    1 2 5 3 6 14

    0 0 1 0 72

    60 0 0 0 1 2

    ⎤⎥⎦ The first (and only) row in the new

    submatrix was multiplied by 2 tointroduce a leading 1.

    The entire matrix is now in row echelon form. To find the reduced row echelon form weneed the following additional step.

    Step 6. Beginning with the last nonzero row and working upward, add suitable multiplesof each row to the rows above to introduce zeros above the leading 1’s.⎡

    ⎢⎣1 2 −5 3 6 140 0 1 0 0 10 0 0 0 1 2

    ⎤⎥⎦ 72 times the third row of the preceding

    matrix was added to the second row.

    ⎡⎢⎣1 2 −5 3 0 20 0 1 0 0 1

    0 0 0 0 1 2

    ⎤⎥⎦ −6 times the third row was added to the

    first row.

    ⎡⎢⎣1 2 0 3 0 70 0 1 0 0 1

    0 0 0 0 1 2

    ⎤⎥⎦ 5 times the second row was added to the

    first row.

    The last matrix is in reduced row echelon form.The procedure (or algorithm) we have just described for reducing a matrix to reduced

    row echelon form is called Gauss–Jordan elimination. This algorithm consists of twoparts, a forward phase in which zeros are introduced below the leading 1’s and a backwardphase in which zeros are introduced above the leading 1’s. If only theforward phase is

    Carl Friedrich Gauss(1777–1855)

    Wilhelm Jordan(1842–1899)

    Historical Note Although versions of Gaussian elimination were known muchearlier, its importance in scientific computation became clear when the greatGerman mathematician Carl Friedrich Gauss used it to help compute the orbitof the asteroid Ceres from limited data. What happened was this: On January 1,1801 the Sicilian astronomer and Catholic priest Giuseppe Piazzi (1746–1826)noticed a dim celestial object that he believed might be a “missing planet.” Henamed the object Ceres and made a limited number of positional observationsbut then lost the object as it neared the Sun. Gauss, then only 24 years old,undertook the problem of computing the orbit of Ceres from the limited datausing a technique called “least squares,” the equations of which he solved bythe method that we now call “Gaussian elimination.” The work of Gauss cre-ated a sensation when Ceres reappeared a year later in the constellation Virgoat almost the precise position that he predicted! The basic idea of the methodwas further popularized by the German engineer Wilhelm Jordan in his bookon geodesy (the science of measuring Earth shapes) entitled Handbuch derVer-messungskunde and published in 1888.

    [Images: Photo Inc/Photo Researchers/Getty Images (Gauss);Leemage/Universal Images Group/Getty Images (Jordan)]

  • 16 Chapter 1 Systems of Linear Equations and Matrices

    used, then the procedure produces a row echelon form and is called Gaussian elimination.For example, in the preceding computations a row echelon form was obtained at the endof Step 5.

    EXAMPLE 5 Gauss–Jordan Elimination

    Solve by Gauss–Jordan elimination.

    x1 + 3x2 − 2x3 + 2x5 = 02x1 + 6x2 − 5x3 − 2x4 + 4x5 − 3x6 = −1

    5x3 + 10x4 + 15x6 = 52x1 + 6x2 + 8x4 + 4x5 + 18x6 = 6

    Solution The augmented matrix for the system is⎡⎢⎢⎢⎣

    1 3 −2 0 2 0 02 6 −5 −2 4 −3 −10 0 5 10 0 15 5

    2 6 0 8 4 18 6

    ⎤⎥⎥⎥⎦

    Adding −2 times the first row to the second and fourth rows gives⎡⎢⎢⎢⎣

    1 3 −2 0 2 0 00 0 −1 −2 0 −3 −10 0 5 10 0 15 5

    0 0 4 8 0 18 6

    ⎤⎥⎥⎥⎦

    Multiplying the second row by −1 and then adding −5 times the new second row to thethird row and −4 times the new second row to the fourth row gives⎡

    ⎢⎢⎢⎣1 3 −2 0 2 0 00 0 1 2 0 3 1

    0 0 0 0 0 0 0

    0 0 0 0 0 6 2

    ⎤⎥⎥⎥⎦

    Interchanging the third and fourth rows and then multiplying the third row of the re-sulting matrix by 16 gives the row echelon form⎡

    ⎢⎢⎢⎣1 3 −2 0 2 0 00 0 1 2 0 3 1

    0 0 0 0 0 1 130 0 0 0 0 0 0

    ⎤⎥⎥⎥⎦ This completes the forward phase sincethere are zeros below the leading 1’s.

    Adding −3 times the third row to the second row and then adding 2 times the secondrow of the resulting matrix to the first row yields the reduced row echelon form⎡

    ⎢⎢⎢⎣1 3 0 4 2 0 0

    0 0 1 2 0 0 0

    0 0 0 0 0 1 130 0 0 0 0 0 0

    ⎤⎥⎥⎥⎦ This completes the backward phase sincethere are zeros above the leading 1’s.

    The corresponding system of equations isNote that in constructing thelinear system in (3) we ignoredthe row of zeros in the corre-sponding augmented matrix.Why is this justified?

    x1 + 3x2 + 4x4 + 2x5 = 0x3 + 2x4 = 0

    x6 = 13(3)

  • 1.2 Gaussian Elimination 17

    Solving for the leading variables, we obtain

    x1 = −3x2 − 4x4 − 2x5x3 = −2x4x6 = 13

    Finally, we express the general solution of the system parametrically by assigning thefree variables x2, x4, and x5 arbitrary values r, s, and t , respectively. This yields

    x1 = −3r − 4s − 2t, x2 = r, x3 = −2s, x4 = s, x5 = t, x6 = 13

    Homogeneous LinearSystems

    A system of linear equations is said to be homogeneous if the constant terms are all zero;that is, the system has the form

    a11x1 + a12x2 + · · ·+ a1nxn = 0a21x1 + a22x2 + · · ·+ a2nxn = 0

    ......

    ......

    am1x1 + am2x2 + · · ·+ amnxn = 0Every homogeneous system of linear equations is consistent because all such systemshave x1 = 0, x2 = 0, . . . , xn = 0 as a solution. This solution is called the trivial solution;if there are other solutions, they are called nontrivial solutions.

    Because a homogeneous linear system always has the trivial solution, there are onlytwo possibilities for its solutions:

    • The system has only the trivial solution.

    • The system has infinitely many solutions in addition to the trivial solution.

    In the special case of a homogeneous linear system of two equations in two unknowns,say

    a1x + b1y = 0 (a1, b1 not both zero)a2x + b2y = 0 (a2, b2 not both zero)

    the graphs of the equations are lines through the origin, and the trivial solution corre-sponds to the point of intersection at the origin (Figure 1.2.1).

    Figure 1.2.1

    x

    y

    Only the trivial solution

    x

    y

    Infinitely manysolutions

    a1x + b1y = 0

    a1x + b1y = 0and

    a2x + b2y = 0

    a2x + b2y = 0

    There is one case in which a homogeneous system is assured of having nontrivialsolutions—namely, whenever the system involves more unknowns than equations. Tosee why, consider the following example of four equations in six unknowns.

  • 18 Chapter 1 Systems of Linear Equations and Matrices

    EXAMPLE 6 A Homogeneous System

    Use Gauss–Jordan elimination to solve the homogeneous linear system

    x1 + 3x2 − 2x3 + 2x5 = 02x1 + 6x2 − 5x3 − 2x4 + 4x5 − 3x6 = 0

    5x3 + 10x4 + 15x6 = 02x1 + 6x2 + 8x4 + 4x5 + 18x6 = 0

    (4)

    Solution Observe first that the coefficients of the unknowns in this system are the sameas those in Example 5; that is, the two systems differ only in the constants on the rightside. The augmented matrix for the given homogeneous system is⎡

    ⎢⎢⎢⎣1 3 −2 0 2 0 02 6 −5 −2 4 −3 00 0 5 10 0 15 02 6 0 8 4 18 0

    ⎤⎥⎥⎥⎦ (5)

    which is the same as the augmented matrix for the system in Example 5, except for zerosin the last column. Thus, the reduced row echelon form of this matrix will be the sameas that of the augmented matrix in Example 5, except for the last column. However,a moment’s reflection will make it evident that a column of zeros is not changed by anelementary row operation, so the reduced row echelon form of (5) is⎡

    ⎢⎢⎢⎣1 3 0 4 2 0 0

    0 0 1 2 0 0 0

    0 0 0 0 0 1 0

    0 0 0 0 0 0 0

    ⎤⎥⎥⎥⎦ (6)

    The corresponding system of equations is

    x1 + 3x2 + 4x4 + 2x5 = 0x3 + 2x4 = 0

    x6 = 0Solving for the leading variables, we obtain

    x1 = −3x2 − 4x4 − 2x5x3 = −2x4x6 = 0

    (7)

    If we now assign the free variables x2, x4, and x5 arbitrary values r , s, and t , respectively,then we can express the solution set parametrically as

    x1 = −3r − 4s − 2t, x2 = r, x3 = −2s, x4 = s, x5 = t, x6 = 0Note that the trivial solution results when r = s = t = 0.

    FreeVariables inHomogeneous Linear

    Systems

    Example 6 illustrates two important points about solving homogeneous linear systems:

    1. Elementary row operations do not alter columns of zeros in a matrix, so the reducedrow echelon form of the augmented matrix for a homogeneous linear system hasa final column of zeros. This implies that the linear system corresponding to thereduced row echelon form is homogeneous, just like the original system.

  • 1.2 Gaussian Elimination 19

    2. When we constructed the homogeneous linear system corresponding to augmentedmatrix (6), we ignored the row of zeros because the corresponding equation

    0x1 + 0x2 + 0x3 + 0x4 + 0x5 + 0x6 = 0does not impose any conditions on the unknowns. Thus, depending on whether ornot the reduced row echelon form of the augmented matrix for a homogeneous linearsystem has any rows of zero, the linear system corresponding to that reduced rowechelon form will either have the same number of equations as the original systemor it will have fewer.

    Now consider a general homogeneous linear system with n unknowns, and supposethat the reduced row echelon form of the augmented matrix has r nonzero rows. Sinceeach nonzero row has a leading 1, and since each leading 1 corresponds to a leadingvariable, the homogeneous system corresponding to the reduced row echelon form ofthe augmented matrix must have r leading variables and n − r free variables. Thus, thissystem is of the form

    xk1 +∑

    ( ) = 0xk2 +

    ∑( ) = 0

    . . ....

    xkr +∑

    ( ) = 0

    (8)

    where in each equation the expression∑

    ( ) denotes a sum that involves the free variables,if any [see (7), for example]. In summary, we have the following result.

    THEOREM 1.2.1 FreeVariableTheorem for Homogeneous Systems

    If a homogeneous linear system has n unknowns, and if the reduced row echelon formof its augmented matrix has r nonzero rows, then the system has n − r free variables.

    Theorem 1.2.1 has an important implication for homogeneous linear systems withNote that Theorem 1.2.2 ap-plies only to homogeneoussystems—a nonhomogeneoussystem with more unknownsthan equations need not beconsistent. However, we willprove later that if a nonho-mogeneous system with moreunknowns then equations isconsistent, then it has in-finitely many solutions.

    more unknowns than equations. Specifically, if a homogeneous linear system has mequations in n unknowns, and if m < n, then it must also be true that r < n (why?).This being the case, the theorem implies that there is at least one free variable, and thisimplies that the system has infinitely many solutions. Thus, we have the following result.

    THEOREM 1.2.2 A homogeneous linear system with more unknowns than equations hasinfinitely many solutions.

    In retrospect, we could have anticipated that the homogeneous system in Example 6would have infinitely many solutions since it has four equations in six unknowns.

    Gaussian Elimination andBack-Substitution

    For small linear systems that are solved by hand (such as most of those in this text),Gauss–Jordan elimination (reduction to reduced row echelon form) is a good procedureto use. However, for large linear systems that require a computer solution, it is generallymore efficient to use Gaussian elimination (reduction to row echelon form) followed bya technique known as back-substitution to complete the process of solving the system.The next example illustrates this technique.

  • 20 Chapter 1 Systems of Linear Equations and Matrices

    EXAMPLE 7 Example 5 Solved by Back-Substitution

    From the computations in Example 5, a row echelon form of the augmented matrix is⎡⎢⎢⎢⎣

    1 3 −2 0 2 0 00 0 1 2 0 3 1

    0 0 0 0 0 1 130 0 0 0 0 0 0

    ⎤⎥⎥⎥⎦

    To solve the corresponding system of equations

    x1 + 3x2 − 2x3 + 2x5 = 0x3 + 2x4 + 3x6 = 1

    x6 = 13we proceed as follows:

    Step 1. Solve the equations for the leading variables.

    x1 = −3x2 + 2x3 − 2x5x3 = 1 − 2x4 − 3x6x6 = 13

    Step 2. Beginning with the bottom equation and working upward, successively substituteeach equation into all the equations above it.

    Substituting x6 = 13 into the second equation yieldsx1 = −3x2 + 2x3 − 2x5x3 = −2x4x6 = 13

    Substituting x3 = −2x4 into the first equation yieldsx1 = −3x2 − 4x4 − 2x5x3 = −2x4x6 = 13

    Step 3. Assign arbitrary values to the free variables, if any.

    If we now assign x2, x4, and x5 the arbitrary values r , s, and t , respectively, thegeneral solution is given by the formulas

    x1 = −3r − 4s − 2t, x2 = r, x3 = −2s, x4 = s, x5 = t, x6 = 13This agrees with the solution obtained in Example 5.

    EXAMPLE 8

    Suppose that the matrices below are augmented matrices for linear systems in the un-knowns x1, x2, x3, and x4. These matrices are all in row echelon form but not reduced rowechelon form. Discuss the existence and uniqueness of solutions to the correspondinglinear systems

  • 1.2 Gaussian Elimination 21

    (a)

    ⎡⎢⎢⎢⎣

    1 −3 7 2 50 1 2 −4 10 0 1 6 9

    0 0 0 0 1

    ⎤⎥⎥⎥⎦ (b)

    ⎡⎢⎢⎢⎣

    1 −3 7 2 50 1 2 −4 10 0 1 6 9

    0 0 0 0 0

    ⎤⎥⎥⎥⎦ (c)

    ⎡⎢⎢⎢⎣

    1 −3 7 2 50 1 2 −4 10 0 1 6 9

    0 0 0 1 0

    ⎤⎥⎥⎥⎦

    Solution (a) The last row corresponds to the equation

    0x1 + 0x2 + 0x3 + 0x4 = 1from which it is evident that the system is inconsistent.

    Solution (b) The last row corresponds to the equation

    0x1 + 0x2 + 0x3 + 0x4 = 0which has no effect on the solution set. In the remaining three equations the variablesx1, x2, and x3 correspond to leading 1’s and hence are leading variables. The variable x4is a free variable. With a little algebra, the leading variables can be expressed in termsof the free variable, and the free variable can be assigned an arbitrary value. Thus, thesystem must have infinitely many solutions.

    Solution (c) The last row corresponds to the equation

    x4 = 0which gives us a numerical value for x4. If we substitute this value into the third equation,namely,

    x3 + 6x4 = 9we obtain x3 = 9. You should now be able to see that if we continue this process andsubstitute the known values of x3 and x4 into the equation corresponding to the secondrow, we will obtain a unique numerical value for x2; and if, finally, we substitute theknown values of x4, x3, and x2 into the equation corresponding to the first row, we willproduce a unique numerical value for x1. Thus, the system has a unique solution.

    Some Facts About EchelonForms

    There are three facts about row echelon forms and reduced row echelon forms that areimportant to know but we will not prove:

    1. Every matrix has a unique reduced row echelon form; that is, regardless of whetheryou use Gauss–Jordan elimination or some other sequence of elementary row oper-ations, the same reduced row echelon form will result in the end.*

    2. Row echelon forms are not unique; that is, different sequences of elementary rowoperations can result in different row echelon forms.

    3. Although row echelon forms are not unique, the reduced row echelon form and allrow echelon forms of a matrix A have the same number of zero rows, and the leading1’s always occur in the same positions. Those are called the pivot positions of A. Acolumn that contains a pivot position is called a pivot column of A.

    *A proof of this result can be found in the article “The Reduced Row Echelon Form of a Matrix Is Unique: ASimple Proof,” by Thomas Yuster, Mathematics Magazine, Vol. 57, No. 2, 1984, pp. 93–94.

  • 22 Chapter 1 Systems of Linear Equations and Matrices

    EXAMPLE 9 Pivot Positions and Columns

    Earlier in this section (immediately after Definition 1) we found a row echelon form of

    A =⎡⎢⎣0 0 −2 0 7 122 4 −10 6 12 28

    2 4 −5 6 −5 −1

    ⎤⎥⎦

    to be ⎡⎢⎣1 2 −5 3 6 140 0 1 0 − 72 −6

    0 0 0 0 1 2

    ⎤⎥⎦

    The leading 1’s occur in positions (row 1, column 1), (row 2, column 3), and (row 3,column 5). These are the pivot positions. The pivot columns are columns 1, 3, and 5.

    If A is the augmented ma-trix for a linear system, thenthe pivot columns identify theleading variables. As an illus-tration, in Example 5 the pivotcolumns are 1, 3, and 6, andthe leading variables arex1, x3,and x6.

    Roundoff Error andInstability

    There is often a gap between mathematical theory and its practical implementation—Gauss–Jordan elimination and Gaussian elimination being good examples. The problemis that computers generally approximate numbers, thereby introducing roundoff errors,so unless precautions are taken, successive calculations may degrade an answer to adegree that makes it useless. Algorithms (procedures) in which this happens are calledunstable. There are various techniques for minimizing roundoff error and instability.For example, it can be shown that for large linear systems Gauss–Jordan eliminationinvolves roughly 50% more operations than Gaussian elimination, so most computeralgorithms are based on the latter method. Some of these matters will be considered inChapter 9.

    Exercise Set 1.2In Exercises 1–2, determine whether the matrix is in row ech-

    elon form, reduced row echelon form, both, or neither.

    1. (a)

    ⎡⎢⎣1 0 00 1 0

    0 0 1

    ⎤⎥⎦ (b)

    ⎡⎢⎣1 0 00 1 0

    0 0 0

    ⎤⎥⎦ (c)

    ⎡⎢⎣0 1 00 0 1

    0 0 0

    ⎤⎥⎦

    (d)

    [1 0 3 1

    0 1 2 4

    ](e)

    ⎡⎢⎢⎢⎣

    1 2 0 3 0

    0 0 1 1 0

    0 0 0 0 1

    0 0 0 0 0

    ⎤⎥⎥⎥⎦

    (f )

    ⎡⎢⎣0 00 0

    0 0

    ⎤⎥⎦ (g)

    [1 −7 5 50 1 3 2

    ]

    2. (a)

    ⎡⎢⎣1 2 00 1 0

    0 0 0

    ⎤⎥⎦ (b)

    ⎡⎢⎣1 0 00 1 0

    0 2 0

    ⎤⎥⎦ (c)

    ⎡⎢⎣1 3 40 0 1

    0 0 0

    ⎤⎥⎦

    (d)

    ⎡⎢⎣1 5 −30 1 1

    0 0 0

    ⎤⎥⎦ (e)

    ⎡⎢⎣1 2 30 0 0

    0 0 1

    ⎤⎥⎦

    (f )

    ⎡⎢⎢⎢⎣

    1 2 3 4 5

    1 0 7 1 3

    0 0 0 0 1

    0 0 0 0 0

    ⎤⎥⎥⎥⎦ (g)

    [1 −2 0 10 0 1 −2

    ]

    In Exercises 3–4, suppose that the augmented matrix for a lin-ear system has been reduced by row operations to the given rowechelon form. Solve the system.

    3. (a)

    ⎡⎢⎣1 −3 4 70 1 2 2

    0 0 1 5

    ⎤⎥⎦

    (b)

    ⎡⎢⎣1 0 8 −5 60 1 4 −9 3

    0 0 1 1 2

    ⎤⎥⎦

    (c)

    ⎡⎢⎢⎢⎣

    1 7 −2 0 −8 −30 0 1 1 6 5

    0 0 0 1 3 9

    0 0 0 0 0 0

    ⎤⎥⎥⎥⎦

    (d)

    ⎡⎢⎣1 −3 7 10 1 4 0

    0 0 0 1

    ⎤⎥⎦

  • 1.2 Gaussian Elimination 23

    4. (a)

    ⎡⎢⎣1 0 0 −30 1 0 0

    0 0 1 7

    ⎤⎥⎦

    (b)

    ⎡⎢⎣1 0 0 −7 80 1 0 3 2

    0 0 1 1 −5

    ⎤⎥⎦

    (c)

    ⎡⎢⎢⎢⎣

    1 −6 0 0 3 −20 0 1 0 4 7

    0 0 0 1 5 8

    0 0 0 0 0 0

    ⎤⎥⎥⎥⎦

    (d)

    ⎡⎢⎣1 −3 0 00 0 1 0

    0 0 0 1

    ⎤⎥⎦

    In Exercises 5–8, solve the linear system by Gaussian elimi-nation.

    5. x1 + x2 + 2x3 = 8−x1 − 2x2 + 3x3 = 13x1 − 7x2 + 4x3 = 10

    6. 2x1 + 2x2 + 2x3 = 0−2x1 + 5x2 + 2x3 = 1

    8x1 + x2 + 4x3 = −17. x − y + 2z − w = −1

    2x + y − 2z − 2w = −2−x + 2y − 4z + w = 13x − 3w = −3

    8. − 2b + 3c = 13a + 6b − 3c = −26a + 6b + 3c = 5In Exercises 9–12, solve the linear system by Gauss–Jordan

    elimination.

    9. Exercise 5 10. Exercise 6

    11. Exercise 7 12. Exercise 8

    In Exercises 13–14, determine whether the homogeneous sys-tem has nontrivial solutions by inspection (without pencil andpaper).

    13. 2x1 − 3x2 + 4x3 − x4 = 07x1 + x2 − 8x3 + 9x4 = 02x1 + 8x2 + x3 − x4 = 0

    14. x1 + 3x2 − x3 = 0x2 − 8x3 = 0

    4x3 = 0In Exercises 15–22, solve the given linear system by any

    method.

    15. 2x1 + x2 + 3x3 = 0x1 + 2x2 = 0

    x2 + x3 = 0

    16. 2x − y − 3z = 0−x + 2y − 3z = 0

    x + y + 4z = 0

    17. 3x1 + x2 + x3 + x4 = 05x1 − x2 + x3 − x4 = 0

    18. v + 3w − 2x = 02u + v − 4w + 3x = 02u + 3v + 2w − x = 0

    −4u − 3v + 5w − 4x = 019. 2x + 2y + 4z = 0

    w − y − 3z = 02w + 3x + y + z = 0

    −2w + x + 3y − 2z = 020. x1 + 3x2 + x4 = 0

    x1 + 4x2 + 2x3 = 0− 2x2 − 2x3 − x4 = 0

    2x1 − 4x2 + x3 + x4 = 0x1 − 2x2 − x3 + x4 = 0

    21. 2I1 − I2 + 3I3 + 4I4 = 9I1 − 2I3 + 7I4 = 11

    3I1 − 3I2 + I3 + 5I4 = 82I1 + I2 + 4I3 + 4I4 = 10

    22. Z3 + Z4 + Z5 = 0−Z1 − Z2 + 2Z3 − 3Z4 + Z5 = 0

    Z1 + Z2 − 2Z3 − Z5 = 02Z1 + 2Z2 − Z3 + Z5 = 0

    In each part of Exercises 23–24, the augmented matrix for alinear system is given in which the asterisk represents an unspec-ified real number. Determine whether the system is consistent,and if so whether the solution is unique. Answer “inconclusive” ifthere is not enough information to make a decision.

    23. (a)

    ⎡⎣1 ∗ ∗ ∗0 1 ∗ ∗

    0 0 1 ∗

    ⎤⎦ (b)

    ⎡⎣1 ∗ ∗ ∗0 1 ∗ ∗

    0 0 0 0

    ⎤⎦

    (c)

    ⎡⎣1 ∗ ∗ ∗0 1 ∗ ∗

    0 0 0 1

    ⎤⎦ (d)

    ⎡⎣1 ∗ ∗ ∗0 0 ∗ 0

    0 0 1 ∗

    ⎤⎦

    24. (a)

    ⎡⎣1 ∗ ∗ ∗0 1 ∗ ∗

    0 0 1 1

    ⎤⎦ (b)

    ⎡⎣1 0 0 ∗∗ 1 0 ∗∗ ∗ 1 ∗

    ⎤⎦

    (c)

    ⎡⎣1 0 0 01 0 0 1

    1 ∗ ∗ ∗

    ⎤⎦ (d)

    ⎡⎣1 ∗ ∗ ∗1 0 0 1

    1 0 0 1

    ⎤⎦

    In Exercises 25–26, determine the values of a for which thesystem has no solutions, exactly one solution, or infinitely manysolutions.

    25. x + 2y − 3z = 43x − y + 5z = 24x + y + (a2 − 14)z = a + 2

  • 24 Chapter 1 Systems of Linear Equations and Matrices

    26. x + 2y + z = 22x − 2y + 3z = 1x + 2y − (a2 − 3)z = a

    In Exercises 27–28, what condition, if any, must a, b, and csatisfy for the linear system to be consistent?

    27. x + 3y − z = ax + y + 2z = b

    2y − 3z = c

    28. x + 3y + z = a−x − 2y + z = b3x + 7y − z = c

    In Exercises 29–30, solve the following systems, where a, b,and c are constants.

    29. 2x + y = a3x + 6y = b

    30. x1 + x2 + x3 = a2x1 + 2x3 = b

    3x2 + 3x3 = c31. Find two different row echelon forms of[

    1 3

    2 7

    ]

    This exercise shows that a matrix can have multiple row eche-lon forms.

    32. Reduce ⎡⎢⎣2 1 30 −2 −29

    3 4 5

    ⎤⎥⎦

    to reduced row echelon form without introducing fractions atany intermediate stage.

    33. Show that the following nonlinear system has 18 solutions if0 ≤ α ≤ 2π , 0 ≤ β ≤ 2π , and 0 ≤ γ ≤ 2π .

    sin α + 2 cos β + 3 tan γ = 02 sin α + 5 cos β + 3 tan γ = 0− sin α − 5 cos β + 5 tan γ = 0

    [Hint: Begin by making the substitutions x = sin α,y = cos β, and z = tan γ .]

    34. Solve the following system of nonlinear equations for the un-known angles α, β, and γ , where 0 ≤ α ≤ 2π , 0 ≤ β ≤ 2π ,and 0 ≤ γ < π .

    2 sin α − cos β + 3 tan γ = 34 sin α + 2 cos β − 2 tan γ = 26 sin α − 3 cos β + tan γ = 9

    35. Solve the following system of nonlinear equations for x, y,and z.

    x2 + y2 + z2 = 6x2 − y2 + 2z2 = 2

    2x2 + y2 − z2 = 3[Hint: Begin by making the substitutions X = x2, Y = y2,Z = z2.]

    36. Solve the following system for x, y, and z.

    1

    x+ 2

    y− 4

    z= 1

    2

    x+ 3

    y+ 8

    z= 0

    − 1x

    + 9y

    + 10z

    = 5

    37. Find the coefficients a, b, c, and d so that the curve shownin the accompanying figure is the graph of the equationy = ax3 + bx2 + cx + d.

    y

    x

    –2 6

    –20

    20(0, 10) (1, 7)

    (3, –11) (4, –14)

    Figure Ex-37

    38. Find the coefficients a, b, c, and d so that the circle shown inthe accompanying figure is given by the equationax2 + ay2 + bx + cy + d = 0.

    y

    x

    (–2, 7)

    (4, –3)

    (–4, 5)

    Figure Ex-38

    39. If the linear system

    a1x + b1y + c1z = 0a2x − b2y + c2z = 0a3x + b3y − c3z = 0

    has only the trivial solution, what can be said about the solu-tions of the following system?

    a1x + b1y + c1z = 3a2x − b2y + c2z = 7a3x + b3y − c3z = 11

    40. (a) If A is a matrix with three rows and five columns, thenwhat is the maximum possible number of leading 1’s in itsreduced row echelon form?

    (b) If B is a matrix with three rows and six columns, thenwhat is the maximum possible number of parameters inthe general solution of the linear system with augmentedmatrix B?

    (c) If C is a matrix with five rows and three columns, thenwhat is the minimum possible number of rows of zeros inany row echelon form of C?

  • 1.3 Matrices and Matrix Operations 25

    41. Describe all possible reduced row echelon forms of

    (a)

    ⎡⎢⎣a b cd e f

    g h i

    ⎤⎥⎦ (b)

    ⎡⎢⎢⎢⎣

    a b c d

    e f g h

    i j k l

    m n p q

    ⎤⎥⎥⎥⎦

    42. Consider the system of equations

    ax + by = 0cx + dy = 0ex + fy = 0

    Discuss the relative positions of the lines ax + by = 0,cx + dy = 0, and ex + fy = 0 when the system has only thetrivial solution and when it has nontrivial solutions.

    Working with Proofs

    43. (a) Prove that if ad − bc �= 0, then the reduced row echelonform of [

    a b

    c d

    ]is

    [1 0

    0 1

    ]

    (b) Use the result in part (a) to prove that if ad − bc �= 0, thenthe linear system

    ax + by = kcx + dy = l

    has exactly one solution.

    True-False Exercises

    TF. In parts (a)–(i) determine whether the statement is true orfalse, and justify your answer.

    (a) If a matrix is in reduced row echelon form, then it is also inrow echelon form.

    (b) If an elementary row operation is applied to a matrix that isin row echelon form, the resulting matrix will still be in rowechelon form.

    (c) Every matrix has a unique row echelon form.

    (d) A homogeneous linear system in n unknowns whose corre-sponding au