INSTRUCTOR’S
SOLUTIONS
MANUAL
THOMAS POLASKI
Winthrop University
JUDITH MCDONALD
Washington State University
L INEAR A LGEBRA AND
I TS APPLICATIONS F OURTH
E DITION
David C. Lay
University of Maryland
,The author and publisher of this book have used their best efforts in preparing this book. These efforts include the
development, research, and testing of the theories and programs to determine their effectiveness. The author and
publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation
contained in this book. The author and publisher shall not be liable in any event for incidental or consequential
damages in connection with, or arising out of, the furnishing, performance, or use of these programs.
Reproduced by Pearson Addison-Wesley from electronic files supplied by the author.
Copyright © 2012, 2006, 1997 Pearson Education, Inc.
Publishing as Pearson Addison-Wesley, 75 Arlington Street, Boston, MA 02116.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any
form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written
permission of the publisher. Printed in the United States of America.
ISBN-13: 978-0-321-38888-9
ISBN-10: 0-321-38888-7
1 2 3 4 5 6 BB 15 14 13 12 11
,Contents
CHAPTER 1 Linear Equations in Linear Algebra 1
CHAPTER 2 Matrix Algebra 87
CHAPTER 3 Determinants 167
CHAPTER 4 Vector Spaces 197
CHAPTER 5 Eigenvalues and Eigenvectors 273
CHAPTER 6 Orthogonality and Least Squares 357
CHAPTER 7 Symmetric Matrices and Quadratic Forms 405
CHAPTER 8 The Geometry of Vector Spaces 453
iii
, 1.1 SOLUTIONS
Notes: The key exercises are 7 (or 11 or 12), 19–22, and 25. For brevity, the symbols R1, R2,…, stand
for row 1 (or equation 1), row 2 (or equation 2), and so on. Additional notes are at the end of the section.
x 5x 7 1 7
1. 2x 1 7 x 2 5 5
5
2 7
1 2
x1 5x2 7 1 5 7
Replace R2 by R2 + (2)R1 and obtain: 9
3x2 9 0 3
Scale R2 by 1/3: x1 5x2 7 1 5 7
0 3
x2 3 1
x1 8 1 0 8
Replace R1 by R1 + (–5)R2: 0 3
x2 3 1
The solution is (x1, x2) = (–8, 3), or simply (–8, 3).
3x1 6x2 3 3 6 3
2. 5x 7x 10 5 7 10
1 2
x1 2x2 1 1 2 1
Scale R1 by 1/3 and obtain: 5 7 10
5x 7 x 10
xReplace R1 by R1 +1 (–2)R2:
1 2x 2 1 2 1
15
3x 15 0 3
Replace R2 by R2 + (–5)R1:
x1 2x2 1 1 2 1
x 5 0 1 5
Scale R2 by –1/3:
x1 9 1 0 9
Copyright © 2012 Pearson Education, Inc. Publishing as Addison-Wesley.
SOLUTIONS
MANUAL
THOMAS POLASKI
Winthrop University
JUDITH MCDONALD
Washington State University
L INEAR A LGEBRA AND
I TS APPLICATIONS F OURTH
E DITION
David C. Lay
University of Maryland
,The author and publisher of this book have used their best efforts in preparing this book. These efforts include the
development, research, and testing of the theories and programs to determine their effectiveness. The author and
publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation
contained in this book. The author and publisher shall not be liable in any event for incidental or consequential
damages in connection with, or arising out of, the furnishing, performance, or use of these programs.
Reproduced by Pearson Addison-Wesley from electronic files supplied by the author.
Copyright © 2012, 2006, 1997 Pearson Education, Inc.
Publishing as Pearson Addison-Wesley, 75 Arlington Street, Boston, MA 02116.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any
form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written
permission of the publisher. Printed in the United States of America.
ISBN-13: 978-0-321-38888-9
ISBN-10: 0-321-38888-7
1 2 3 4 5 6 BB 15 14 13 12 11
,Contents
CHAPTER 1 Linear Equations in Linear Algebra 1
CHAPTER 2 Matrix Algebra 87
CHAPTER 3 Determinants 167
CHAPTER 4 Vector Spaces 197
CHAPTER 5 Eigenvalues and Eigenvectors 273
CHAPTER 6 Orthogonality and Least Squares 357
CHAPTER 7 Symmetric Matrices and Quadratic Forms 405
CHAPTER 8 The Geometry of Vector Spaces 453
iii
, 1.1 SOLUTIONS
Notes: The key exercises are 7 (or 11 or 12), 19–22, and 25. For brevity, the symbols R1, R2,…, stand
for row 1 (or equation 1), row 2 (or equation 2), and so on. Additional notes are at the end of the section.
x 5x 7 1 7
1. 2x 1 7 x 2 5 5
5
2 7
1 2
x1 5x2 7 1 5 7
Replace R2 by R2 + (2)R1 and obtain: 9
3x2 9 0 3
Scale R2 by 1/3: x1 5x2 7 1 5 7
0 3
x2 3 1
x1 8 1 0 8
Replace R1 by R1 + (–5)R2: 0 3
x2 3 1
The solution is (x1, x2) = (–8, 3), or simply (–8, 3).
3x1 6x2 3 3 6 3
2. 5x 7x 10 5 7 10
1 2
x1 2x2 1 1 2 1
Scale R1 by 1/3 and obtain: 5 7 10
5x 7 x 10
xReplace R1 by R1 +1 (–2)R2:
1 2x 2 1 2 1
15
3x 15 0 3
Replace R2 by R2 + (–5)R1:
x1 2x2 1 1 2 1
x 5 0 1 5
Scale R2 by –1/3:
x1 9 1 0 9
Copyright © 2012 Pearson Education, Inc. Publishing as Addison-Wesley.
