Copyr

ight

ed M

ater

ial

BrownW

alke

rPre

ss.c

om

A BRIEF COURSE IN LINEAR ALGEBRA

Copyr

ight

ed M

ater

ial

BrownW

alke

rPre

ss.c

om

Copyr

ight

ed M

ater

ial

BrownW

alke

rPre

ss.c

om

A BRIEF COURSE IN LINEAR ALGEBRA:

MATRICES AND MATRIX EQUATIONS FOR UNDERGRADUATE STUDENTS IN APPLIED

MATHEMATICS, SCIENCE AND ENGINEERING

WHYE-TEONG ANG

BrownWalker Press Irvine & Boca Raton

Copyr

ight

ed M

ater

ial

BrownW

alke

rPre

ss.c

om

A Brief Course in Linear Algebra: Matrices and Matrix Equations for Undergraduate Students in Applied Mathematics, Science and Engineering

Copyright © 2019 Whye-Teong Ang. All rights reserved. No part of this

publication may be reproduced, distributed, or transmitted in any form or by any means, including photocopying, recording, or other electronic or

mechanical methods, without the prior written permission of the publisher, except in the case of brief quotations embodied in critical reviews and certain

other noncommercial uses permitted by copyright law.

BrownWalker Press • Universal Publishers, Inc. Irvine, California & Boca Raton, Florida • USA

www.BrownWalkerPress.com 2019

ISBN: 978-1-62734-724-2 (pbk.) ISBN: 978-1-62734-725-9 (ebk.)

Cover design by Ivan Popov

Publisher's Cataloging-in-Publication Data Names: Ang, Whye-Teong, author. Title: A brief course in linear algebra : matrices and matrix equations for

undergraduate students in applied mathematics, science and engineering / Whye-Teong Ang.

Description: Irvine, CA : BrownWalker, 2019. Identifiers: LCCN 2019939671 | ISBN 978-1-62734-724-2 (paperback) |

ISBN 978-1-62734-725-9 (ebook) Subjects: LCSH: Algebras, Linear. | Algebras, Linear--Study and teaching

(Higher) | Matrices--Study and teaching (Higher) | Algebra. | Mathemat-ics. | BISAC: MATHEMATICS / Algebra / Linear. | MATHEMATICS / Study & Teaching.

Classification: LCC QA184.2 .A54 2019 (print) | LCC QA184.2 (ebook) | DDC 512/.5--dc23.

Copyr

ight

ed M

ater

ial

BrownW

alke

rPre

ss.c

om

Preface

This book contains a brief course in elementary linear alge-

bra with an emphasis on solving systems of linear algebraic and

ordinary differential equations. It is written for undergraduate

students in the disciplines of applied mathematics, science and en-

gineering. Basic knowledge of the arithmetic of complex numbers

and exposure to elementary functions and calculus are assumed.

The book comprises six chapters.

Chapter 1 covers the basics of matrices and vectors, providing

definitions and concepts needed in linear algebra studies in later

chapters.

Chapter 2 is concerned with solving systems of linear alge-

braic equations. It shows how elementary row operations on an

array of numbers can be used to reduce a given system of lin-

ear algebraic equations to a simpler but equivalent system that

can be easily solved. The chapter also introduces the concept of

linearly independent vectors and explains how the task of deter-

mining whether a given set of vectors is linearly independent or

not can be formulated in terms of a homogeneous system of linear

algebraic equations.

Chapter 3 looks at elementary matrices and matrix inverses.

It shows how elementary row operations can be performed on an

invertible square matrix to find its inverse matrix and explains

how matrix invertibility is related to solving a system of linear al-

gebraic equations. Formulae for some properties involving inverses

of matrices are given in the chapter.

Chapter 4 begins with a formula defining the determinant of

a square matrix, shows how elementary row operations can be

performed on a square matrix to calculate its determinant, and

derives alternative formulae for calculating the determinant. The

relation between matrix determinant, matrix inverse and solutions

of systems of linear algebraic equations is explained.

Chapter 5 deals with the matrix eigenproblem and the matrix

diagonalization problem. The two related problems are of funda-

mental importance in linear algebra. The chapter explains how

v

Copyr

ight

ed M

ater

ial

BrownW

alke

rPre

ss.c

om

they can be applied to solve homogeneous systems of first order

linear ordinary differential equations.

Chapter 6 gives a summary of the definition of terms and the

main results in the earlier chapters.

The connections between the topics covered are carefully elu-

cidated. Derivations or proofs are given for all the main results

studied.

Problems are set at the end of each of the first five chapters to

test the understanding of students and to provide further insights

into the topics covered in the course.

W. T. ANGSingapore 10 June 2019

vi

Copyr

ight

ed M

ater

ial

BrownW

alke

rPre

ss.c

om

Contents

1 Basics of matrices and vectors 11

1.1 Definition and notation . . . . . . . . . . . . . . . 11

1.1.1 Null matrices . . . . . . . . . . . . . . . . . 12

1.1.2 Submatrices . . . . . . . . . . . . . . . . . . 12

1.2 Square matrices . . . . . . . . . . . . . . . . . . . . 13

1.2.1 Upper triangular matrices . . . . . . . . . . 13

1.2.2 Lower triangular matrices . . . . . . . . . . 14

1.2.3 Diagonal matrices . . . . . . . . . . . . . . 15

1.2.4 Identity matrices . . . . . . . . . . . . . . . 15

1.3 Arithmetic of matrices . . . . . . . . . . . . . . . . 15

1.3.1 Equality of matrices . . . . . . . . . . . . . 15

1.3.2 Addition of matrices . . . . . . . . . . . . . 16

1.3.3 Multiplication of a number to a matrix . . 17

1.3.4 Product of matrices . . . . . . . . . . . . . 17

1.3.5 Powers of square matrices . . . . . . . . . . 21

1.4 Transpose of a matrix . . . . . . . . . . . . . . . . 23

1.5 Vectors . . . . . . . . . . . . . . . . . . . . . . . . 25

1.5.1 Ordered sets of numbers . . . . . . . . . . . 25

1.5.2 Euclidean norm of a vector . . . . . . . . . 26

1.5.3 Inner product of vectors . . . . . . . . . . . 26

1.5.4 Orthogonal vectors . . . . . . . . . . . . . . 27

1.5.5 Arithmetic of vectors . . . . . . . . . . . . . 27

1.5.6 Linear combination of vectors . . . . . . . . 29

1.5.7 Real vector spaces and subspaces . . . . . . 32

1.6 Problems . . . . . . . . . . . . . . . . . . . . . . . 35

2 Linear algebraic equations and linearly independent

vectors 39

2.1 Linear algebraic equations . . . . . . . . . . . . . . 39

vii

Copyr

ight

ed M

ater

ial

BrownW

alke

rPre

ss.c

om

2.2 Solving systems of linear algebraic equations . . . . 41

2.2.1 Systems of linear algebraic equations . . . . 41

2.2.2 Solution of a system . . . . . . . . . . . . . 43

2.2.3 Elimination of unknowns . . . . . . . . . . 44

2.2.4 Elementary row operations . . . . . . . . . 47

2.2.5 General observations . . . . . . . . . . . . . 58

2.3 Linearly independent vectors . . . . . . . . . . . . 61

2.3.1 Basic definition . . . . . . . . . . . . . . . . 61

2.3.2 Homogeneous systems with infinitely many

solutions . . . . . . . . . . . . . . . . . . . . 67

2.4 Problems . . . . . . . . . . . . . . . . . . . . . . . 69

3 Inverse of a square matrix 73

3.1 Elementary row operations and elementary matrices 73

3.2 Invertible square matrices . . . . . . . . . . . . . . 79

3.2.1 Inverse of a matrix . . . . . . . . . . . . . . 79

3.2.2 Inverses of elementary matrices . . . . . . . 80

3.2.3 Finding the inverse of a matrix by row op-

erations . . . . . . . . . . . . . . . . . . . . 83

3.2.4 Invertible matrices as products of elemen-

tary matrices . . . . . . . . . . . . . . . . . 88

3.2.5 Some properties involving inverses of matrices 91

3.3 Invertible matrices and systems of linear algebraic

equations . . . . . . . . . . . . . . . . . . . . . . . 93

3.4 Elementary column row operations . . . . . . . . . 94

3.5 Problems . . . . . . . . . . . . . . . . . . . . . . . 96

4 Determinant of a square matrix 99

4.1 Starting definition for determinant . . . . . . . . . 99

4.2 Determinant and elementary row and column oper-

ations . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.2.1 Effects of elementary row operations on de-

terminant . . . . . . . . . . . . . . . . . . . 104

4.2.2 Effects of elementary column operations on

determinant . . . . . . . . . . . . . . . . . 106

4.2.3 Evaluating determinant by row operations . 109

4.3 Some properties of determinant . . . . . . . . . . . 115

4.3.1 Product of square matrices . . . . . . . . . 115

4.3.2 Inverse of a square matrix . . . . . . . . . . 117

4.3.3 Transpose of a square matrix . . . . . . . . 117

viii

Copyr

ight

ed M

ater

ial

BrownW

alke

rPre

ss.c

om

4.4 Alternative equivalent definition for determinant . 119

4.5 Determinant in terms of determinants of submatrices119

4.6 Cofactor, adjugate and inverse . . . . . . . . . . . 125

4.7 Determinant and systems of linear algebraic equations128

4.8 Problems . . . . . . . . . . . . . . . . . . . . . . . 132

5 Eigenvalues and eigenvectors, diagonalization and

differential equations 135

5.1 Matrix eigenproblem . . . . . . . . . . . . . . . . . 135

5.2 Eigenvalues and eigenvectors . . . . . . . . . . . . 138

5.3 Linearly independent eigenvectors . . . . . . . . . . 146

5.4 Matrix diagonalization problem . . . . . . . . . . . 149

5.5 Diagonalization matrices . . . . . . . . . . . . . . . 152

5.6 Real symmetric matrices . . . . . . . . . . . . . . . 157

5.6.1 Eigenvalues of a real symmetric matrix . . 157

5.6.2 Eigenvectors of a real symmetric matrix . . 157

5.6.3 Diagonalizing real symmetric matrices . . . 158

5.7 Solutions of homogeneous linear ordinary differen-

tial equations with constant coefficients . . . . . . 168

5.7.1 Systems of first order homogeneous linear

ordinary differential equations with constant

coefficients . . . . . . . . . . . . . . . . . . 168

5.7.2 Solutions of first order systems with diago-

nalizable coefficient matrices . . . . . . . . 169

5.7.3 Solutions of first order systems with nondi-

agonalizable coefficient matrices . . . . . . 176

5.8 Problems . . . . . . . . . . . . . . . . . . . . . . . 185

6 Summary 189

6.1 Definitions and concepts . . . . . . . . . . . . . . . 189

6.2 Main results . . . . . . . . . . . . . . . . . . . . . . 194

ix

Copyr

ight

ed M

ater

ial

BrownW

alke

rPre

ss.c

om

Copyr

ight

ed M

ater

ial

BrownW

alke

rPre

ss.c

om

Chapter 1

Basics of matrices and

vectors

1.1 Definition and notation

A matrix of order× , that is, an× matrix, is a collection

(set) of numbers arranged in rows and columns. For

clarity, a pair of brackets is used to enclose the numbers in a

matrix.

Examples:

1. (10) (1× 1 matrix)

2.

µ2 −90 1

¶(2× 2 matrix)

3.

⎛⎜⎜⎜⎜⎝1 −1 22 5 210 2 −101 1 −9√2 1

⎞⎟⎟⎟⎟⎠ (5× 3 matrix)

4.

⎛⎝ 3 6 4 9110 1 12 12 2 4 5

⎞⎠ (3× 4 matrix).

11

Copyr

ight

ed M

ater

ial

BrownW

alke

rPre

ss.c

om

We refer to the numbers in a matrix as the elements of the

matrix. The elements of a matrix may be real or complex numbers.

We use bold capital letters such asABM andN to represent

matrices.

Example:

If A and B are used to represent respectively the 5× 3 and 3× 4matrices in the examples given above, then we write:

A =

⎛⎜⎜⎜⎜⎝1 −1 22 5 210 2 −101 1 −9√2 1

⎞⎟⎟⎟⎟⎠ and B =

⎛⎝ 3 6 4 9110 1 12 12 2 4 5

⎞⎠

If we denote the element in the -th row and the -th columnof an × matrix A by then we can write:

A = () =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

11 12 13 · · · · · · 121 22 23 · · · · · · 231 32 33 · · · · · · 3...

......

. . ....

......

.... . .

...

1 2 3 · · · · · ·

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

1.1.1 Null matrices

If all the elements in a matrix are zero, the matrix is a null (or

zero) matrix. The × null matrix is denoted by O× If theorder of a null matrix is implicitly understood in a discussion, the

null matrix may be simply written as O

1.1.2 Submatrices

A matrix formed by deleting away selected rows and/or columns

of a larger matrix A is called a submatrix of AA matrix can be partitioned and expressed in terms of its sub-

matrices.

12

Copyr

ight

ed M

ater

ial

BrownW

alke

rPre

ss.c

om

Examples:

1. If we delete away the first row, the third row and the last

two columns of the 4 × 4 matrix A =

⎛⎜⎜⎝1 3 8 22 3 6 43 0 7 01 0 5 9

⎞⎟⎟⎠ we

obtain

µ2 31 0

¶which is a 2× 2 submatrix of A

2. If we define P =

⎛⎝ 1 6 5 93 1 8 12 7 4 6

⎞⎠ R =

⎛⎝ 1 63 12 7

⎞⎠ and

S =

⎛⎝ 5 98 14 6

⎞⎠ we can express the 3× 4 matrix P in terms

of its 3 × 2 submatrices R and S as P =¡R S

¢. The

3 × 4 matrix ¡ S R¢, the 6 × 2 matrix

µRS

¶and the

6× 4 matrixµRO3×2

O3×2S

¶are given by

⎛⎝ 5 98 14 6

1 63 12 7

⎞⎠,⎛⎜⎜⎜⎜⎜⎜⎝

1 63 12 75 98 14 6

⎞⎟⎟⎟⎟⎟⎟⎠ and

⎛⎜⎜⎜⎜⎜⎜⎝1 63 12 70 00 00 0

0 00 00 05 98 14 6

⎞⎟⎟⎟⎟⎟⎟⎠respectively.

1.2 Square matrices

A matrix is square if its number of rows equals its number of

columns. The elements 11 22 · · · (−1)(−1) and in an

× square matrix A are called the diagonal elements of A

1.2.1 Upper triangular matrices

A square matrix A = () is an upper triangular matrix if = 0for all , that is, if all the elements below the diagonal elementsare zero.

13

Copyr

ight

ed M

ater

ial

BrownW

alke

rPre

ss.c

om

Examples:

The matrices listed below are upper triangular.

1.

µ2 90 6

¶(diagonal elements are 2 and 6)

2.

⎛⎝ 1 1 00 5 20 0 3

⎞⎠ (diagonal elements are 1 5 and 3)

3.

⎛⎜⎜⎝1 5 3 20 3 2 10 0 0 60 0 0 4

⎞⎟⎟⎠ (diagonal elements are 1 3 0 and 4)

1.2.2 Lower triangular matrices

A square matrix A = () is a lower triangular matrix if = 0for all , that is, if all the elements above the diagonal elementsare zero.

Examples:

The matrices listed below are lower triangular.

1.

µ1 05 2

¶

2.

⎛⎝ 1 0 08 5 06 1 0

⎞⎠

3.

⎛⎜⎜⎝5 0 0 01 3 0 00 1 8 01 0 9 4

⎞⎟⎟⎠

14

Copyr

ight

ed M

ater

ial

BrownW

alke

rPre

ss.c

om

1.2.3 Diagonal matrices

If all the nondiagonal elements of a square matrixA are zero, then

A is a diagonal matrix. A diagonal matrix is both lower and uppertriangular.

Examples:

1.

⎛⎝ 1 0 00 1 00 0 6

⎞⎠ (3× 3 diagonal matrix)

2.

⎛⎜⎜⎜⎜⎝−10 0 0 0 00 0 0 0 00 0 12 0 00 0 0 1 00 0 0 0 2

⎞⎟⎟⎟⎟⎠ (5× 5 diagonal matrix)

1.2.4 Identity matrices

A diagonal matrix is said to be an identity matrix if all the diagonal

elements of the matrix is 1 The × identity matrix is denoted

by I× (which may be simply written as I if the order of thematrix is understood).

Example:

I5×5 =

⎛⎜⎜⎜⎜⎝1 0 0 0 00 1 0 0 00 0 1 0 00 0 0 1 00 0 0 0 1

⎞⎟⎟⎟⎟⎠

1.3 Arithmetic of matrices

1.3.1 Equality of matrices

Two matrices A = () and B = () both of order × aresaid to be equal, that is, A = B if = , that is, if the element

15

Copyr

ight

ed M

ater

ial

BrownW

alke

rPre

ss.c

om

in the -th row and -th column of A is equal to the element in

the -th row and -th column of B.

Example:

If the matrices

µ+ −

¶and

µ 3

¶are equal to each

other, find and

Solution:

Since µ+ −

¶=

µ 3

¶

we can equate the corresponding elements in the two matrices to

obtain + = = 3 = and − = Thus, = 2, = 1 = 3 and = 3

1.3.2 Addition of matrices

Two matrices A = () and B = () both of order × canbe added up to form another × matrix denoted by A +BIf S = () = A+B then = +

Example:⎛⎝ 1 23 45 6

⎞⎠+⎛⎝ −1 2

8 30 6

⎞⎠=

⎛⎝ 1− 1 2 + 23 + 8 4 + 35 + 0 6 + 6

⎞⎠ =

⎛⎝ 0 411 75 12

⎞⎠It is obvious that A+B = B+A, that is, addition of matrices

is commutative, and A +O = O +A = A where O is the null

matrix of the same order as AIf C = () is another matrix of order × then

(A+B) +C = A+ (B+C)

since ( + ) + = + ( + ) Note that the usual ruleof evaluating first the mathematical expression enclosed within a

pair of brackets is also applicable in matrix operations.

16

Copyr

ight

ed M

ater

ial

BrownW

alke

rPre

ss.c

om

1.3.3 Multiplication of a number to a matrix

If A = () is an × matrix and is a number then A is an

× matrix defined by A = () where = We write(−1)A as simply −A and define B−A = B+ (−A) (if A and Bare both ×)

Examples:

1. 2

µ32

¶=

µ64

¶

2. (−1)µ32

¶= −

µ32

¶=

µ −3−2

¶

3.

µ85

¶−µ12

¶=

µ73

¶

It is obvious that A+ A = (+ )A for any numbers and and any matrix A Also, 0A = O and A−A = O

1.3.4 Product of matrices

LetA = () be an× matrix and B = () an × matrix.We can form the product AB, which is an × matrix, only if

= . If = and AB = M = () then the element in the-th row and -th column of M, that is, is defined in termsof the elements in the -th row of A and the -th column of B by

= 11 + 22 + · · ·+

=X=1

that is, is the sum of the products of all the correspond-

ing pairs of elements in the submatrices¡1 2 · · ·

¢(the -th row of A) and

⎛⎜⎜⎜⎝12...

⎞⎟⎟⎟⎠ (the -th column of B) Cor-

responding pairs of elements are formed by taking the element in

17

Copyr

ight

ed M

ater

ial

BrownW

alke

rPre

ss.c

om

the -th column of¡1 2 · · ·

¢and that in the -th

row of

⎛⎜⎜⎜⎝12...

⎞⎟⎟⎟⎠ For example, 3 and 3 form a corresponding

pair of elements.

For given integer values and , where 1 ≤ ≤ and

1 ≤ ≤ if we regard ( ) as a 1× 1 submatrix of AB, wemay write:

( ) =¡1 2 · · ·

¢⎛⎜⎜⎜⎝

12...

⎞⎟⎟⎟⎠= (11 + 22 + · · ·+ )

From the above definition of the product of matrices, it is clear

that if is a number then (AB) = (A)B = A(B) Hence theproducts (AB) (A)B and A(B) are the same matrix and maybe simply written as AB

Examples:

1. Since R =

⎛⎝ 1 23 45 6

⎞⎠ and S =

µ5 1 2 23 3 1 2

¶are of order

3 × 2 and 2 × 4 respectively, we can form the product RSwhich is a 3×4 matrix. The element in the -th row and the-th column ofRS can be calculated by using the elements inthe -th row ofR and the -th column of S. For example, theelement in the second row and third column of RS is given

by the element in the 1 × 1 submatrix ¡ 3 4¢µ 2

1

¶=

(6 + 4) = (10) If we calculate all the elements of of RS, weobtain

RS =

⎛⎝ 1 23 45 6

⎞⎠µ 5 1 2 23 3 1 2

¶=

⎛⎝ 11 7 4 627 15 10 1443 23 16 22

⎞⎠

18

Copyr

ight

ed M

ater

ial

BrownW

alke

rPre

ss.c

om

In this example, we cannot form the product SR (that is,

SR does not exist), since the number of columns of S is notequal to the number of rows of R

2. If P =

⎛⎝ 1 36 45 6

⎞⎠ andQ =

µ7 1 23 2 1

¶ both PQ andQP

exist and can be worked out as follows:

PQ =

⎛⎝ 1 36 45 6

⎞⎠µ 7 1 23 2 1

¶=

⎛⎝ 16 7 554 14 1653 17 16

⎞⎠

QP =

µ7 1 23 2 1

¶⎛⎝ 1 36 45 6

⎞⎠ =

µ23 3720 23

¶

The products PQ and QP cannot be equal to each other,

as the order of PQ is different from that of QP.

3. If R =

µ1 36 4

¶and S =

µ7 13 2

¶ we find that

RS =

µ1 36 4

¶µ7 13 2

¶=

µ16 754 14

¶

SR =

µ7 13 2

¶µ1 36 4

¶=

µ13 2515 17

¶6= RS

For general matrices A and B such that AB and BA ex-

ist, AB needs not necessarily be equal to BA, even if ABand BA have the same order. For some matrices, we may

possibly find that AB = BA In general, multiplication ofmatrices is, however, not commutative.

4. If U =

µ73

¶and V =

¡10 1

¢then

I2×2U =

µ1 00 1

¶µ73

¶=

µ73

¶= U

VI2×2 =¡10 1

¢µ 1 00 1

¶=¡10 1

¢= V

19

Copyr

ight

ed M

ater

ial

BrownW

alke

rPre

ss.c

om

If the elements in the identity matrix I× are denoted by then

=

½1 if = 0 if 6=

If A = () is an× matrix and AI× = C = () then

= 11 + 22 + · · ·+

Depending on the value of all the terms, except one, in thesum 11 + 22 + · · · + are zero. If = 1 then1 = 1 Similarly, if = 2 then 2 = 2 In general, we findthat = that is, C = A Thus, if the product AI exists,then AI = A

Also, if I×A = D = () then

= 11 + 22 + · · ·+ =

Thus, if IA exists then IA = ALetA = () be an× matrix andB = () andC = ()

be × matrices. IfW = () = A(B+C) then we can write

=X=1

( + )

=X=1

+X=1

Note thatP=1

andP=1

are the elements in the -th

row and the -th column of the product matrices AB and ACrespectively. Thus, we find thatW = AB+AC that is,

A(B+C) = AB+AC

Similarly, if A is an × matrix and B and C are ×matrices, we can show that

(B+C)A = BA+CA

Note the order of appearance of A in the matrix products on both

sides of each of the formulae above. The order is important as

multiplication of matrices is, in general, not commutative.

20

Copyr

ight

ed M

ater

ial

BrownW

alke

rPre

ss.c

om

Let A = () B = () and C = () be matrices of order × × and × respectively. Define the × matrix

F = () = AB where

=X=1

If the product FC = (AB)C is given by the × matrix G =() then

=X

=1

=X

=1

(X=1

) =X=1

(X

=1

)

Note thatP

=1 gives the element in the -th row and -

th column of the product matrix BC Hence we can write G =A(BC) that is,

(AB)C = A(BC)

Multiplication of matrices is associative. Since (AB)C andA(BC)are equal to each other, they can just be written as ABC Notethat ABC 6= ACB as multiplication of matrices is not commuta-tive.

1.3.5 Powers of square matrices

Let A be an × matrix.

We define:

A1 = A

A2 = AA1 = AA

A3 = AA2 = AAA

A4 = AA3 = AAAA...

A = AA−1 = AAA · · ·A| {z }product of matrices

...

21

Copyr

ight

ed M

ater

ial

BrownW

alke

rPre

ss.c

om

Note that A is of order × and the matrices A2 A3A4 andso on do not exist if A is not a square matrix.

It is obvious that if is a number then (A) = AIf A and B are square matrices of the same order then

(A+B)2 = (A+B)(A+B)

= (A+B)A+ (A+B)B

= A2 +BA+AB+B2

Note that we cannot write (A+B)2 = A2 + 2AB+B2 or (A+B)2 = A2 + 2BA+B2 because BA 6= AB in general.

If the diagonal element on the -th row of an × diagonal

matrix D is then the matrix D ( is a positive integer) is

also an × diagonal matrix with diagonal element on the

-th row

Example:⎛⎝ −1 0 00 2 00 0 3

⎞⎠6 =⎛⎝ (−1)6 0 0

0 26 00 0 36

⎞⎠ =

⎛⎝ 1 0 00 64 00 0 729

⎞⎠We can prove the proposition that D is diagonal with diag-

onal element on the -th row by using the principle of mathe-matical induction.

The proposition is true for = 1 since D1 = DWe shall now show that if the proposition is true for =

( is a given positive integer) then it is also true for = +1Let D = () where

=

½ if = = 0 if 6=

Define D = () and D+1 = ()

If the above mentioned proposition is true for = then

=

½ if = = 0 if 6=

22

Copyr

ight

ed M

ater

ial

BrownW

alke

rPre

ss.c

om

Since D+1 = DD , we can write

=X=1

= 11 + 22 + 33 + · · ·+

For any fixed value of if 6= either or (or both) iszero. Hence = 0 if 6= that is, D+1 = () is a diagonalmatrix.

If = = the term is zero for all values of except = It follows that

= = = +1

that is, the diagonal element in the -th row of D+1 is +1

Thus, if the proposition that D is diagonal with diagonal

element on the -th row is true for = it is also truefor = + 1 As pointed out above, the proposition is true for = 1 By induction, we deduce that it is true for = 2 3 4· · ·

1.4 Transpose of a matrix

Let A = () be an × matrix. We define the transpose of

A to be the × matrix B = (), where = for = 12 · · · and = 1 2 · · · The elements in the -th columnof B are the elements in the -th row of A in the same order of

appearance. The transpose of A is denoted by A It should beobvious that (A ) = A and (A) = A where is a number.

Examples:

1.¡1 10 9 3

¢=

⎛⎜⎜⎝11093

⎞⎟⎟⎠

2.

µ23

¶

=¡2 3

¢

23

Copyr

ight

ed M

ater

ial

BrownW

alke

rPre

ss.c

om

3.

⎛⎝ 1 42 30 5

⎞⎠

=

µ1 2 04 3 5

¶

4.

⎛⎜⎝⎛⎝ 1 42 30 5

⎞⎠⎞⎟⎠

=

µ1 2 04 3 5

¶

=

⎛⎝ 1 42 30 5

⎞⎠

5.

⎛⎜⎜⎝1 6 5 02 7 5 74 1 8 25 4 9 8

⎞⎟⎟⎠

=

⎛⎜⎜⎝1 2 4 56 7 1 45 5 8 90 7 2 8

⎞⎟⎟⎠

6.

µ2

µ1 62 7

¶¶

=

µ2 412 14

¶= 2

µ1 62 7

¶

Let V = () andW = () be matrices of order × . IfP = () = V+W P = () V

= () andW = () then

= + for = 1 2 · · · and = 1 2 · · ·

and

= + = +

for = 1 2 · · · and = 1 2 · · ·

From = + we know that P = V+W that is,

(V+W)= V +W

Let V = () andW = () be two matrices of order ×and × respectively. If P = () = VW P = () V

=(),W

= () and Z = () =WV then

=X=1

for = 1 2 · · · and = 1 2 · · ·

24

Copyr

ight

ed M

ater

ial

BrownW

alke

rPre

ss.c

om

and

=X=1

=X=1

=X=1

= =

for = 1 2 · · · and = 1 2 · · ·

From = we know that P =WV that is,

(VW) =WV

1.5 Vectors

1.5.1 Ordered sets of numbers

In linear algebra, an -dimensional vector is an ordered set of numbers. A vector is defined not only by the numbers in the set

but also by their order of appearance in the set.

An -dimensional vector may be represented by either a 1×matrix or an × 1 matrix. We use underlined bold letters suchas u v and w to denote vectors. Thus, if u is a four-dimensionalvector, we may represent it by using a 1 × 4 matrix or a 4 × 1matrix, that is, we may write

u =¡1 2 3 4

¢or u =

⎛⎜⎜⎝1234

⎞⎟⎟⎠

We refer to -dimensional vectors represented by 1 × ma-

trices and × 1 matrices as row and column vectors respectively.The numbers in the ordered sets defining vectors are referred to

as the vector components. The vector components of real vectors

are restricted to real numbers. In general, the vector components

may be complex.

In elementary physics, vectors are physical quantities in two or

three-dimensional space, having magnitudes and directions. Ex-

amples of such vector quantities are displacement and velocity.

The displacement of a body from a point is characterized by the

distance and the direction of the body from the point. The dis-

tance which is given by a number is the magnitude of the displace-

ment. The velocity of the body is characterized by the speed at

25