P. Sivaramakrishna Das, C. Vijayakumari - Engineering Mathematics-Pearson Education (2017).pdf

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About Pearson
A01_ENGINEERING_MATHEMATICS-I _FM - (Reprint).indd 1 3/2/2017 6:17:51 PM

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Engineering Mathematics
P. Sivaramakrishna Das
Professor of Mathematics
and
Head of the P.G. Department of Mathematics (Retired)
Ramakrishna Mission Vivekananda College
Mylapore, Chennai
Presently
Professor of Mathematics
and
Head of the Department of Science and Humanities
K.C.G College of Technology
(a unit of Hindustan Group of Institutions
Karapakkam, Chennai)
C. Vijayakumari
Professor of Mathematics (Retired)
Queen Mary’s College (Autonomous)
Mylapore, Chennai

Copyright © 2017 Pearson India Education Services Pvt. Ltd
Published by Pearson India Education Services Pvt. Ltd, CIN: U72200TN2005PTC057128, formerly
known as TutorVista Global Pvt. Ltd, licensee of Pearson Education in South Asia.
No part of this eBook may be used or reproduced in any manner whatsoever without the publisher’s
prior written consent.
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reserves the right to remove any material in this eBook at any time.
eISBN 978-93-325-8776-2
Registered office: 4th Floor, Software Block, Elnet Software City, TS-140, Block 2 & 9, Rajiv
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Noida 201 301, Uttar Pradesh, India.
ISBN 978-93-325-1912-1

Dedicated
to
Our Beloved Parents

Prefacexxxi
About the Authors xxxiii
A. ALGEBRA
1. Matrices 1.1
2. Sequences and Series 2.1
B. CALCULUS
3. Differential Calculus 3.1
4. Applications of Differential Calculus 4.1
5. Differential Calculus of Several Variables 5.1
6. Integral Calculus 6.1
7. Improper Integrals 7.1
8. Multiple Integrals 8.1
9. Vector Calculus 9.1
C. DIFFERENTIAL EQUATIONS
10. Ordinary First Order Differential Equations 10.1
11. Ordinary Second and Higher Order Differential Equations 11.1
12. Applications of Ordinary Differential Equations 12.1
13. Series Solution of Ordinary Differential Equations and Special Functions 13.1
14. Partial Differential Equations 14.1
Brief Contents

viii n Brief Contents
D. COMPLEX ANALYSIS
15. Analytic Functions 15.1
16. Complex Integration 16.1
E. SERIES AND TRANSFORMS
17. Fourier Series 17.1
18. Fourier Transforms 18.1
19. Laplace Transforms 19.1
F. APPLICATIONS
20. Applications of Partial Differential Equations 20.1
Index I.1

Prefacexxxi
About the Authors xxxiii
1. Matrices 1.1
1.0 Introduction 1.1
1.1 Basic Concepts 1.1
1.1.1 Basic Operations on Matrices 1.4
1.1.2 Properties of Addition, Scalar Multiplication and Multiplication 1.5
1.2 Complex Matrices 1.7
Worked Examples 1.10
Exercise 1.1 1.13
Answers to Exercise 1.1 1.14
1.3 Rank of a Matrix 1.14
Exercise 1.21.23
Answers to Exercise 1.21.24
1.4 Solution of System of Linear Equations 1.24
1.4.1 Non-homogeneous System of Equations 1.24
1.4.2 Homogeneous System of Equations 1.25
1.4.3 Type 1: Solution of Non-homogeneous System of Equations 1.26
Worked Examples1.26
1.4.4 Type 2: Solution of Non-homogeneous Linear Equations Involving
Arbitrary Constants 1.34
Worked Examples1.34
1.4.5 Type 3: Solution of the System of Homogeneous Equations 1.38
Worked Examples1.38
1.4.6 Type 4: Solution of Homogeneous System of Equation Containing
Arbitrary Constants 1.41
Worked Examples1.41
Exercise 1.31.44
1.5 Matrix Inverse by Gauss–Jordan method 1.46
Worked Examples1.47
Exercise 1.41.53
Contents

x n Contents
1.6 Eigen Values and Eigen Vectors 1.54
1.6.0 Introduction 1.54
1.6.1 Vector 1.54
1.6.2 Eigen Values and Eigen Vectors 1.56
1.6.3 Properties of Eigen Vectors 1.57
1.6.4 Properties of Eigen Values 1.67
Exercise 1.5 1.72
1.6.5 Cayley-Hamilton Theorem 1.73
Exercise 1.6 1.82
1.7 Similarity Transformation and Orthogonal Transformation 1.83
1.7.1 Similar Matrices 1.83
1.7.2 Diagonalisation of a Square Matrix 1.84
1.7.3 Computation of the Powers of a Square Matrix 1.85
1.7.4 Orthogonal matrix 1.86
1.7.5 Properties of orthogonal matrix 1.86
1.7.6 Symmetric Matrix 1.87
1.7.7 Properties of Symmetric Matrices 1.88
1.7.8
Diagonalisation by Orthogonal Transformation or Orthogonal Reduction 1.89
1.8 Real Quadratic Form. Reduction to Canonical Form 1.96
Exercise 1.7 1.111
Short Answer Questions1.113
Objective Type Questions1.114
Answers1.116
2. Sequences and Series 2.1
2.1 Sequence 2.1
2.1.1 Infinite Sequence 2.1
2.1.2 Finite sequence 2.2
2.1.3 Limit of a sequence 2.2
2.1.4 Convergent sequence 2.2
2.1.5 Oscillating sequence 2.2
2.1.6 Bounded sequence 2.2
2.1.7 Monotonic Sequence 2.3
Worked Examples 2.3
Exercise 2.1 2.9

Contents n xi
2.2 Series 2.9
2.2.1 Convergent Series 2.9
2.2.2 Divergent series2.10
2.2.3 Oscillatory series2.10
2.2.4 General properties of series2.10
2.3 Series of Positive Terms 2.10
2.3.1 Necessary Condition for Convergence of a Series 2.10
2.3.2 Test for convergence of positive term series2.11
2.3.3 Comparison tests2.11
Worked Examples2.13
Exercise 2.22.17
2.3.4 De’ Alembert’s Ratio Test 2.18
Worked Examples2.21
Exercise 2.32.25
2.3.5 Cauchy’s Root Test 2.27
Worked Examples2.28
2.3.6 Cauchy’s Integral Test 2.30
Worked Examples2.32
Exercise 2.42.36
2.3.7 Raabe’s Test 2.36
Worked Examples2.37
Exercise 2.52.41
2.3.8 Logarithmic Test 2.42
Worked Examples2.44
2.4 Alternating Series 2.46
2.4.1 Leibnitz’s Test 2.46
Worked Examples2.47
2.5 Series of Positive and Negative Terms 2.50
2.5.1 Absolute Convergence and Conditional Convergence 2.50
2.5.2 Tests for absolute convergence2.50
Worked Examples2.51
Exercise 2.62.55
2.6 Convergence of Binomial Series 2.56
2.7 Convergence of the Exponential Series 2.57
2.8 Convergence of the Logarithmic Series 2.57
2.9 Power Series 2.58
2.9.1 Hadmard’s Formula 2.59
2.9.2 Properties of power series2.60
Worked Examples2.60
Exercise 2.72.66

xii n Contents
Short Answer Questions 2.67
Objective Type Questions 2.69
Answers 2.70
3. Differential Calculus 3.1
3.1 Successive Differentiation 3.2
Worked Examples 3.3
Exercise 3.1 3.6
3.1.1 The nth
Derivative of Standard Functions 3.7
Exercise 3.2 3.16
Exercise 3.33.24
3.2 Applications of Derivative 3.25
3.2.1 Geometrical Interpretation of Derivative 3.25
3.2.2 Equation of the Tangent and the Normal to the Curve y = f(x)3.25
Worked Examples3.26
Exercise 3.43.33
3.2.3 Length of the Tangent, the Sub-Tangent, the Normal and the Sub-normal 3.34
Worked Examples3.36
Exercise 3.53.38
3.2.4 Angle between the Two Curves 3.38
Worked Examples3.39
Exercise 3.63.42
3.3 Mean-value Theorems of Derivatives 3.43
3.3.1 Rolle’s Theorem 3.43
Worked Examples3.44
3.3.2 Lagrange’s Mean Value Theorem 3.47
Worked Examples3.49
3.3.3 Cauchy’s Mean Value Theorem 3.53
Worked Examples3.54
Exercise 3.73.56
3.4 Monotonic Functions 3.58
3.4.1 Increasing and Decreasing Functions 3.58
3.4.2 Piece-wise Monotonic Function 3.58
3.4.3 Test for increasing or decreasing functions3.59
Worked Examples3.60
Exercise 3.83.65

Contents n xiii
3.5 Generalised Mean Value Theorem 3.66
3.5.1 Taylor’s Theorem with Lagrange’s Form of Remainder 3.66
3.5.2 Taylor’s series 3.68
3.5.3 Maclaurin’s theorem with Lagrange’s form of remainder 3.68
3.5.4 Maclaurin’s series 3.68
Exercise 3.9 3.74
3.5.5 Expansion by Using Maclaurin’s Series of Some Standard Functions 3.75
3.5.6 Expansion of Certain Functions Using Differential Equations 3.78
Exercise 3.10 3.81
3.6 Indeterminate Forms 3.82
3.6.1 General L’Hopital’s Rule for
0
0
Form 3.84
Exercise 3.11 3.94
3.7 Maxima and Minima of a Function of One Variable 3.94
3.7.1 Geometrical Meaning 3.96
3.7.2 Tests for Maxima and Minima 3.96
Summary 3.97
Exercise 3.12 3.103
3.8 Asymptotes 3.104
Worked Examples3.105
3.8.1 A General Method 3.108
3.8.2 Asymptotes Parallel to the Coordinates Axes 3.110
3.8.3 Another Method for Finding the Asymptotes 3.113
3.8.4 Asymptotes by Inspection 3.115
3.8.5 Intersection of a Curve and Its Asymptotes 3.116
Exercise 3.13 3.121
3.9 Concavity 3.122
Exercise 3.143.127
3.10 Curve Tracing 3.128
3.10.1 Procedure for Tracing the Curve Given by the Cartesian
Equation f(x, y) = 0. 3.128

xiv n Contents
3.10.2 Procedure for Tracing of Curve Given by Parametric Equations
x = f(t), y = g(t)3.137
3.10.3 Procedure for Tracing of Curve Given by Equation in Polar
Coordinates f(r, u) = 0 3.141
Exercise 3.153.146
Answers3.152
4. Applications of Differential Calculus 4.1
4.1 Curvature in Cartesian Coordinates 4.1
4.1.0 Introduction 4.1
4.1.1 Measure of Curvature 4.1
4.1.2 Radius of Curvature for Cartesian Equation of a Given Curve 4.2
4.1.3 Radius of Curvature for Parametric Equations 4.4
Worked Examples 4.4
4.1.4 Centre of Curvature and Circle of Curvature 4.11
4.1.5 Coordinates of the Centre of Curvature 4.12
Exercise 4.1 4.15
4.1.6 Radius of Curvature in Polar Coordinates 4.17
4.1.7 Radius of Curvature at the Origin 4.22
4.1.8 Pedal Equation or p - r Equation of a Curve 4.25
4.1.9 Radius of Curvature Using the p - r Equation of a Curve 4.28
Exercise 4.2 4.30
4.2 Evolute 4.31
4.2.1 Properties of Evolute 4.31
4.2.2 Procedure to Find the Evolute 4.34
Exercise 4.3 4.41
4.3 Envelope 4.42
4.3.1
Method of Finding Envelope of Single Parameter Family of Curves 4.42
4.3.2 Envelope of Two Parameter Family of Curves 4.45

Contents n xv
4.3.3 Evolute as the Envelope of Normals 4.48
Worked Examples4.49
Exercise 4.44.52
Answers4.56
5. Differential Calculus of Several Variables 5.1
5.1 Limit and Continuity 5.1
Worked Examples 5.4
Exercise 5.1 5.6
5.2 Partial Derivatives 5.6
5.2.1 Geometrical Meaning of
∂
∂
∂
∂
z
x
z
y
, 5.7
5.2.2 Partial Derivatives of Higher Order 5.8
5.2.3 Homogeneous Functions and Euler’s Theorem 5.8
Worked Examples 5.9
5.2.4 Total Derivatives 5.15
Exercise 5.25.24
5.3 Jacobians 5.26
5.3.1 Properties of Jacobians 5.27
Worked Examples5.29
5.3.2 Jacobian of Implicit Functions 5.35
Worked Examples5.35
Exercise 5.35.37
5.4
Taylor’s Series Expansion for Function of Two Variables 5.38
Worked Examples5.39
Exercise 5.45.44
5.5
Maxima and Minima for Functions of Two Variables 5.45
5.5.1 Necessary Conditions for Maximum or Minimum 5.46
5.5.2 Sufficient Conditions for Extreme Values of f(x, y) 5.46
5.5.3 Working Rule to Find Maxima and Minima of f(x, y) 5.46
Worked Examples5.47
5.5.4 Constrained Maxima and Minima 5.51
5.5.5 Lagrange’s Method of (undetermined) Multiplier 5.51
5.5.6 Method to Decide Maxima or Minima 5.52
Worked Examples5.56
Exercise 5.55.65

xvi n Contents
5.6 Errors and Approximations 5.67
Exercise 5.6 5.72
Objective Type Questions 5.74
Answers 5.76
6. Integral Calculus 6.1
6.1 Indefinite Integral 6.1
6.1.1 Properties of Indefinite Integral 6.1
6.1.2 Integration by Parts 6.3
6.1.3 Bernoulli’s Formula 6.3
6.1.4 Special Integrals 6.3
Worked Examples 6.4
Exercise 6.1 6.9
6.2 Definite Integral (Newton–Leibnitz formula) 6.10
6.2.1 Properties of Definite Integral 6.10
Worked Examples6.15
Exercise 6.26.27
6.3 Definite Integral f x dx
a
b
( )
∫ as Limit of a Sum 6.28
6.3.1 Working Rule 6.28
Worked Examples6.29
Exercise 6.36.32
6.4 Reduction Formulae 6.33
6.4.1 The Reduction Formula for (a) sinn
x dx
∫ and (b) cosn
x dx
∫ 6.33
6.4.2 The Reduction Formula for (a) tann
x dx
∫ and (b) cotn
xdx
∫ 6.36
6.4.3 The Reduction Formula for (a) secn
x dx
∫ and (b) cosecn
x dx
∫ 6.37
Worked Examples6.38
6.4.4
The Reduction Formula for sin cos
m n
x x dx
∫ , Where m, n are
Non-negative Integers6.45
Worked Examples6.47
6.4.5
The Reduction Formula for (a) ∫xm
(log x)n
dx, (b) ∫xn
sin mx dx,
(c) ∫ xn
cos mx dx6.49
6.4.6 The Reduction Formula for (a) e x dx
ax m
sin
∫ and (b) e x dx
ax n
cos
∫ 6.51

Contents n xvii
6.4.7 The Reduction Formula for (a) cos sin
m
x nx dx
∫ and (b) cos cos
m
x nx dx
∫ 6.52
Exercise 6.4 6.55
6.5 Application of Integral Calculus 6.55
6.5.1 Area of Plane Curves 6.56
6.5.1 (a) Area of Plane Curves in Cartesian Coordinates 6.56
Exercise 6.5 6.66
6.5.1 (b) Area in Polar Coordinates 6.67
Exercise 6.6 6.72
6.5.2 Length of the Arc of a Curve 6.72
6.5.2 (a) Length of the Arc in Cartesian Coordinates 6.72
Exercise 6.7 6.78
6.5.2 (b) Length of the Arc in Polar Coordinates 6.79
Exercise 6.8 6.81
6.5.3 Volume of Solid of Revolution 6.82
6.5.3(a) Volume in Cartesian Coordinates 6.82
Exercise 6.9 6.89
6.5.3 (b) Volume in Polar Coordinates 6.91
Exercise 6.10 6.93
6.5.4 Surface Area of Revolution 6.93
6.5.4(a) Surface Area of Revolution in Cartesian Coordinates 6.93
Exercise 6.11 6.99
6.5.4 (b) Surface Area in Polar Coordinates 6.100
Exercise 6.126.102
Answers6.106

xviii n Contents
7. Improper Integrals 7.1
7.1 Improper Integrals 7.1
7.1.1 Kinds of Improper Integrals and Their Convergence 7.1
Worked Examples 7.4
Exercise 7.17.13
7.1.2 Tests of Convergence of Improper Integrals 7.14
Exercise 7.27.27
7.2 Evaluation of Integral by Leibnitz’s Rule 7.27
7.2.1 Leibnitz’s Rule—Differentiation Under Integral Sign for Variable Limits 7.28
Worked Examples7.28
Exercise 7.3 7.47
7.3 Beta and Gamma functions 7.47
7.3.1 Beta Function 7.47
7.3.2 Symmetric property of beta function7.48
7.3.3 Different forms of beta function7.48
7.4 The Gamma Function 7.49
7.4.1 Properties of Gamma Function 7.50
7.4.2 Relation between Beta and Gamma Functions 7.51
Worked Examples7.55
Exercise 7.47.69
7.5 The Error Function 7.70
7.5.1 Properties of Error Functions 7.70
7.5.2 Series expansion for error function7.71
7.5.3 Complementary error function7.71
Worked Examples7.72
Exercise 7.57.76
Answers7.78
8. Multiple Integrals 8.1
8.1 Double Integration 8.1
8.1.1 Double Integrals in Cartesian Coordinates 8.1
8.1.2 Evaluation of Double Integrals 8.2
Worked Examples 8.3
Exercise 8.1 8.6
8.1.3 Change of Order of Integration 8.7
Worked Examples 8.8

Contents n xix
Exercise 8.28.15
8.1.4 Double Integral in Polar Coordinates 8.16
Worked Examples8.16
8.1.5 Change of Variables in Double Integral 8.19
Worked Examples8.19
Exercise 8.38.26
8.1.6 Area as Double Integral 8.27
Worked Examples8.28
Exercise 8.48.31
Worked Examples8.32
Exercise 8.58.37
8.2 Area of a Curved Surface 8.37
8.2.1 Surface Area of a Curved Surface 8.38
8.2.2 Derivation of the Formula for Surface Area 8.38
8.2.3 Parametric Representation of a Surface 8.41
Worked Examples8.41
Exercise 8.68.49
8.3 Triple Integral in Cartesian Coordinates 8.49
Worked Examples8.50
Exercise 8.78.55
8.3.1 Volume as Triple Integral 8.56
Worked Examples8.56
Exercise 8.88.63
Answers8.66
9. Vector Calculus 9.1
9.1 Scalar and Vector Point Functions 9.1
9.1.1 Geometrical Meaning of Derivative 9.2
9.2 Differentiation Formulae 9.3
9.3 Level Surfaces 9.4
9.4
Gradient of a Scalar Point Function or Gradient of a Scalar Field 9.4
9.4.1 Vector Differential Operator 9.4
9.4.2 Geometrical Meaning of ∇φ 9.4
9.4.3 Directional Derivative 9.5
9.4.4 Equation of Tangent Plane and Normal to the Surface 9.5

xx n Contents
9.4.5 Angle between Two Surfaces at a Common Point 9.6
9.4.6 Properties of gradients 9.6
Worked Examples 9.8
Exercise 9.1 9.20
9.5
Divergence of a Vector Point Function or Divergence of a Vector Field 9.22
9.5.1 Physical Interpretation of Divergence 9.22
9.6
Curl of a Vector Point Function or Curl of a Vector Field 9.23
9.6.1 Physical Meaning of Curl F 9.23
Exercise 9.2 9.30
9.7 Vector Identities 9.31
9.8 Integration of Vector Functions 9.39
9.8.1 Line Integral 9.40
Exercise 9.3 9.46
9.9 Green’s Theorem in a Plane 9.47
9.9.1 Vector Form of Green’s Theorem 9.50
9.10 Surface Integrals 9.56
9.10.1 Evaluation of Surface Integral 9.57
9.11 Volume Integral 9.58
9.12 Gauss Divergence Theorem 9.62
9.12.1 Results Derived from Gauss Divergence Theorem 9.64
9.13 Stoke’s Theorem 9.81
Exercise 9.4 9.97
Answers9.102
10. Ordinary First Order Differential Equations 10.1
10.1 Formation of Differential Equations 10.2
Exercise 10.1 10.5
10.2 First Order and First Degree Differential Equations 10.6
10.2.1 Type I
Variable Separable Equations 10.6
=
1
curl
2
w v

Contents n xxi
Worked Example 10.6
Exercise 10.2 10.9
10.2.2 Type II
Homogeneous Equation 10.10
Exercise 10.310.13
10. 2.3 Type III
Non-Homogenous Differential Equations of the First Degree 10.14
Exercise 10.410.21
10.2.4 Type IV
Linear Differential Equation 10.22
Exercise 10.510.27
10.2.5 Type V Bernoulli’s Equation 10.28
Exercise 10.610.31
10.2.6 Type VI Riccati Equation 10.31
Exercise 10.710.36
10.2.7 Type VII First Order Exact Differential Equations 10.37
Exercise 10.810.41
10.3 Integrating Factors 10.42
10.3.1
Rules for Finding the Integrating Factor for Non-Exact Differential
Equation Mdx + Ndy = 0 10.45
Exercise 10.910.56
10.4
Ordinary Differential Equations of the First Order but of Degree Higher
than One 10.56
10.4.1 Type 1 Equations Solvable for p 10.57
Exercise 10.1010.59
10.4.2 Type 2
Equations Solvable for y10.60

xxii n Contents
10.4.3 Type 3
Equations Solvable for x10.64
Exercise 10.1110.67
10.4.4 Type 4 Clairaut’s Equation 10.67
Exercise 10.1210.71
Answers10.74
11. Ordinary Second and Higher Order Differential Equations 11.1
11.1
Linear Differential Equation with Constant Coefficients 11.1
11.1.1 Complementary Function 11.1
11.1.2 Particular Integral 11.2
Exercise 11.111.19
11.2
Linear Differential Equations with Variable Coefficients 11.21
11.2.1 Cauchy’s Homogeneous Linear Differential Equations 11.21
11.2.2 Legendre’s Linear Differential Equation 11.29
Exercise 11.211.32
11.3
Simultaneous Linear Differential Equations with
Constant Coefficients 11.34
Exercise 11.311.43
11.4 Method of Variation of Parameters 11.44
11.4.1 Working rule11.45
Exercise 11.411.51
11.5 Method of Undetermined Coefficients 11.52
Exercise 11.511.60
Short Answers Questions11.60
Answers 11.63

Contents n xxiii
12. Applications of Ordinary Differential Equations 12.1
12.1 Applications of Ordinary Differential Equations of First Order 12.1
12.1.1 Law of Growth and Decay 12.1
12.1.2 Newton’s Law of Cooling of Bodies 12.2
Exercise 12.1 12.7
Answers To Exercise 12.1 12.8
12.1.3 Chemical Reaction and Solutions 12.8
Exercise 12.212.12
12.1.4 Simple Electric Circuit 12.13
Exercise 12.312.19
12.1.5 Geometrical Applications 12.20
12.1.5 (a) Orthogonal Trajectories in Casterian Coordinates 12.20
12.1.5 (b) Orthogonal Trajectories in Polar Coordinates 12.23
Exercise 12.412.26
12.2 Applications of Second Order Differential Equations 12.27
12.2.1 Bending of Beams 12.27
12.2.2 Electric Circuits 12.34
Exercise 12.512.38
12.2.3 Simple Harmonic Motion (S.H.M) 12.40
Exercise 12.612.43
Answers12.45
13. Series Solution of Ordinary Differential Equations and Special Functions 13.1
13.1 Power Series Method 13.1
13.1.1 Analytic Function 13.1
13.1.2 Regular Point 13.2
13.1.3 Singular Point 13.2
13.1.4 Regular and Irregular Singular Points 13.2
Exercise 13.1 13.9

xxiv n Contents
13.2 Frobenius Method 13.9
Exercise 13.213.33
13.3 Special Functions 13.34
13.4 Bessel Functions 13.34
13.4.1 Series Solution of Bessel’s Equation 13.34
13.4.2 Bessel’s Functions of the First Kind 13.37
13.4.3 Some Special Series 13.40
13.4.4 Recurrence Formula for Jn (x)13.41
13.4.5 Generating Function for Jn (x) of Integral Order 13.44
13.4.6 Integral Formula for Bessel’s Function Jn (x) 13.49
13.4.7 Orthogonality of Bessel’s Functions 13.56
13.4.8 Fourier–Bessel Expansion of a Function f(x)13.59
13.4.9 Equations Reducible to Bessel’s Equation 13.62
Exercise 13.313.65
13.5 Legendre Functions 13.66
13.5.1 Series Solution of Legendre’s Differential Equation 13.66
13.5.2 Legendre Polynomials 13.71
13.5.3 Rodrigue’s Formula 13.71
13.5.4 Generating Function for Legendre Polynomials 13.74
13.5.5 Orthogonality of Legendre Polynomials in [-1, 1] 13.77
13.5.6 Fourier–Legendre Expansion of f(x) in a Series of Legendre
Polynomials13.83
Exercise 13.413.85
14. Partial Differential Equations 14.1
14.1 Order and Degree of Partial Differential Equations 14.1
14.2 Linear and Non-linear Partial Differential Equations 14.1
14.3 Formation of Partial Differential Equations 14.2
Exercise 14.114.15

Contents n xxv
14.4 Solutions of Partial Differential Equations 14.16
14.4.1 Procedure to find general integral and singular
integral for a first order partial differential equation14.17
Exercise 14.214.20
14.4.2 First Order Non-linear Partial Differential Equation of Standard Types 14.20
Exercise 14.314.25
14.4.3 Equations Reducible to Standard Forms 14.33
Exercise 14.414.38
14.5 Lagrange’s Linear Equation 14.39
Exercise 14.514.48
14.6 Homogeneous Linear Partial Differential Equations of the Second
and Higher Order with Constant Coefficients 14.49
14.6.1 Working Procedure to Find Complementary Function 14.50
14.6.2 Working Procedure to Find Particular Integral 14.51
Exercise 14.614.66
14.7 Non-homogeneous Linear Partial Differential Equations of the
Second and Higher Order with Constant Coefficients 14.68
Exercise 14.714.73
Answers 14.76
15. Analytic Functions 15.1
15.0 Preliminaries 15.1
15.1 Function of a Complex Variable 15.2
15.1.1 Geometrical Representation of Complex Function or Mapping 15.3
15.1.2 Extended complex number system 15.4
15.1.3 Neighbourhood of a point and region 15.5
15.2 Limit of a Function 15.5
15.2.1 Continuity of a function 15.6
15.2.2 Derivative of f(z) 15.6
15.2.3 Differentiation formulae 15.7

xxvi n Contents
15.3 Analytic Function 15.8
15.3.1
Necessary and Sufficient Condition for f(z)to be Analytic 15.8
15.3.2 C-R equations in polar form15.10
Exercise 15.115.20
15.4 Harmonic Functions and Properties of Analytic Function 15.21
15.4.1
Construction of an Analytic Function Whose Real or
Imaginary Part is Given Milne-Thomson Method 15.23
Exercise 15.215.32
15.5 Conformal Mapping 15.33
15.5.1 Angle of rotation15.34
15.5.2 Mapping by elementary functions15.36
Exercise 15.315.72
15.5.3 Bilinear Transformation 15.79
Exercise 15.415.89
Answers15.92
16. Complex Integration 16.1
16.1 Contour Integral 16.1
16.1.1 Properties of Contour Integrals 16.1
16.1.2 Simply Connected and Multiply Connected Domains 16.4
16.2 Cauchy’s Integral Theorem or Cauchy’s Fundamental Theorem 16.4
16.2.1 Cauchy-Goursat Integral Theorem 16.5
16.3 Cauchy’s Integral Formula 16.6
16.3.1 Cauchy’s Integral Formula for Derivatives 16.7
Exercise 16.116.12
16.4 Taylor’s Series and Laurent’s Series 16.14
16.4.1 Taylor’s Series 16.14
16.4.2 Laurent’s series16.15
Exercise 16.216.22

Contents n xxvii
16.5 Classification of Singularities 16.24
16.6 Residue 16.26
16.6.1 Methods of Finding Residue 16.26
16.7 Cauchy’s Residue Theorem 16.27
Exercise 16.316.34
16.8 Application of Residue Theorem to Evaluate Real Integrals 16.36
16.8.1 Type 1 16.36
16.8.2 Type 2. Improper Integrals of Rational Functions 16.44
16.8.3 Type 3 16.50
Exercise 16.416.55
Answers16.60
17 Fourier Series 17.1
17.1 Fourier series 17.2
17.1.1 Dirichlet’s Conditions 17.2
17.1.2 Convergence of Fourier Series 17.3
17.2 Even and Odd Functions 17.15
17.2.1 Sine and Cosine Series 17.15
Exercise 17.117.23
17.3 Half-Range Series 17.26
17.3.1 Half-range Sine Series 17.27
17.3.2 Half-range cosine series17.27
Exercise 17.217.36
17.4 Change of Interval 17.38
17.5 Parseval’s Identity 17.47
Exercise 17.317.50
17.6 Complex Form of Fourier Series 17.53

xxviii n Contents
Exercise 17.417.59
17.7 Harmonic Analysis 17.60
17.7.1 Trapezoidal Rule 17.60
Exercise 17.517.68
Answers17.72
18. Fourier Transforms 18.1
18.1 Fourier Integral Theorem 18.1
18.1.1 Fourier Cosine and Sine Integrals 18.2
18.1.2 Complex Form of Fourier Integral 18.6
18.2 Fourier Transform Pair 18.7
18.2.1 Properties of Fourier transforms 18.8
Exercise 18.118.21
18.3 Fourier Sine and Cosine Transforms 18.23
18.3.1 Properties of Fourier Sine and Cosine Transforms 18.24
Exercise 18.218.39
18.4 Convolution Theorem 18.40
18.4.1 Definition: Convolution of Two Functions 18.40
18.4.2 Theorem 18.1: Convolution theorem or Faltung theorem18.41
18.4.3 Theorem 18.2 : Parseval’s identity for Fourier transforms or
Energy theorem18.41
Exercise 18.318.51
Answers18.54
19. Laplace Transforms 19.1
19.1
Condition for Existence of Laplace Transform 19.1
19.2
Laplace Transform of Some Elementary Functions 19.2
19.3 Some Properties of Laplace Transform 19.4

Contents n xxix
Exercise 19.1 19.9
19.4
Differentiation and Integration of Transforms 19.11
Exercise 19.219.20
19.5
Laplace Transform of Derivatives and Integrals 19.21
19.5.1
Evaluation of Improper Integrals using Laplace Transform 19.25
19.6
Laplace Transform of Periodic Functions and Other Special Type of Functions 19.27
19.6.1 Laplace Transform of Unit Step Function 19.36
19.6.2 Unit impulse function19.37
19.6.3 Dirac-delta function19.37
19.6.4 Laplace transform of delta function19.37
Exercise 19.319.39
19.7 Inverse Laplace Transforms 19.41
19.7.1 Type 1 – Direct and shifting methods19.43
19.7.2 Type 2 – Partial Fraction Method 19.44
19.7.3 Type 3 – 1. Multiplication by s and 2. Division by s19.48
19.7.4
Type 4 – Inverse Laplace Transform of Logarithmic and
Trigonometric Functions 19.50
Exercise 19.419.53
19.7.5 Type 5 – Method of Convolution 19.55
Exercise 19.519.60
19.7.6 Type 6: Inverse Laplace Transform as Contour Integral 19.61
Exercise 19.619.64
19.8
Application of Laplace Transform to the Solution of Ordinary
Differential Equations 19.65
19.8.1 First Order Linear Differential Equations with Constant Coefficients 19.65
19.8.2
Ordinary Second and Higher Order Linear Differential Equations
with Constant Coefficients 19.68

xxx n Contents
19.8.3 Ordinary Second Order Differential Equations with Variable Coefficients 19.72
Exercise 19.719.75
19.8.4 Simultaneous Differential Equations 19.77
19.8.5 Integral–Differential Equation 19.83
Exercise 19.819.85
Answers19.88
20. Applications of Partial Differential Equations 20.1
20.1 One Dimensional Wave Equation – Equation of Vibrating String 20.2
20.1.1 Derivation of Wave Equation 20.2
20.1.2
Solution of One-Dimensional Wave Equation by the Method of
Separation of Variables (or the Fourier Method) 20.3
Exercise 20.120.34
20.1.3 Classification of Partial Differential Equation of Second Order 20.36
Exercise 20.220.38
20.2 One-Dimensional Equation of Heat Conduction (In a Rod) 20.39
20.2.1 Derivation of Heat Equation 20.39
20.2.2 Solution of Heat Equation by Variable Separable Method 20.40
Exercise 20.320.62
Exercise 20.420.68
20.3 Two Dimensional Heat Equation in Steady State 20.69
20.3.1 Solution of Two Dimensional Heat Equation 20.70
Exercise 20.520.83
Answers20.88
IndexI.1

This book Engineering Mathematics is written to cover the topics that are common to the syllabi of
various universities in India. Although this book is designed primarily for engineering courses, it
is also suitable for Mathematics courses and for various competitive examinations. The aim of the
book is to provide a sound understanding of Mathematics. The experiences of both the authors in
teaching undergraduate and postgraduate students from diverse backgrounds for over four decades
have helped to present the subject as simple as possible with clarity and rigour in a step-by-step
approach.
This book has many distinguishing features. The topics are well organized to create
self-confidence and interest among the readers to study and apply the mathematical tools in
engineering and science disciplines. The subject is presented with a lot of standard worked examples
and exercises that will help the readers to develop maturity in Mathematics.
This book is organized into 20 chapters. At the end of each chapter, short answer questions and
objective questions are given to enhance the understanding of the topics.
Chapter 1 focuses on the applications of matrices to the consistency of simultaneous linear equations
and Eigen value problems.
Chapter 2 discusses convergence of sequence and series.
Chapter 3 deals with differentiation and applications of derivative, Rolle’s Theorem, mean value
theorems, asymptotes and curve tracing.
Chapter 4 deals with the geometrical application of derivative in radius of curvature, centre of
curvature, evolute and envelope.
Chapter 5 elaborates calculus of several variables.
Chapter 6 deals with integral calculus and applications of integral calculus.
Chapter 7 discusses improper integrals, and beta and gamma functions.
Chapter 8 focuses on multiple integrals.
Chapter 9 deals with vector calculus.
Chapter 10 discusses solution of various types of first order differential equations.
Chapter 11 is concerned with the solution of second order and higher order linear differential
equations.
Chapter 12 deals with some applications of ordinary differential equations.
Chapter 13 conforms to series solution of ordinary differential equations and special functions.
Chapter 14 focuses on solution of partial differential equations.
Preface

xxxii n Preface
Chapter 15 examines analytic functions.
Chapter 16 focuses on complex integration.
Chapter 17 deals with Fourier series.
Chapter 18 pertains to Fourier transforms.
Chapter 19 discusses Laplace transforms.
Chapter 20 is concerned with applications of partial differential equations.
Mathematics is a subject that can be mastered only through hard work and practice. Follow the
maximum, Mathematics without practice is blind and practice without understanding is futile.
“Tell me and I will forget
Show me and I will remember
Involve me and I will understand”
—Confucius
We hope that this book is student-friendly and that it will be well received by students and teachers.
We heartily welcome valuable comments and suggestions from our readers for the improvement of
this book, which may be addressed to profpsdas@yahoo.com.
ACKNOWLEDGEMENTS
P. Sivaramakrishna Das: I express my gratitude to our chairperson, Dr Elizabeth Varghese, and
the directors of K.C.G. College of Technology for giving me an opportunity to write this book.
I am obliged to my department colleagues for their encouragement.
The inspiration to write this book came from my wife, Prof. C. Vijayakumari, who is also the
co-author of this book.
P. Sivaramakrishna Das and C. Vijayakumari: We are grateful to the members of our family for
lending us their support for the successful completion of this book.
We are obliged to Sojan Jose, R. Dheepika and C. Purushothaman of Pearson India Education
Services Pvt. Ltd, for their diligence in bringing this work out to fruition.
P. Sivaramakrishna Das
C. Vijayakumari

Prof. Dr P. Sivaramakrishna Das started his career in 1967 as assistant
professor of Mathematics at Ramakrishna Mission Vivekananda College,
Chennai, his alma mater and retired as Head of the P.G. Department of
Mathematics from the same college after an illustrious career spanning
36 years.
Currently, he is professor of Mathematics and Head of the Department
of Science and Humanities, K.C.G. College of Technology, Chennai
(a unit of Hindustan Group of Institutions).
P. Sivaramakrishna Das has done pioneering research work in the
field of “Fuzzy Algebra” and possess a Ph.D. in this field. His paper
on fuzzy groups and level subgroups was a fundamental paper on fuzzy
algebra with over 600 citations and it was the first paper from India. With a teaching experience
spanning over 49 years, he is an accomplished teacher of Mathematics at undergraduate and
postgraduate levels of Arts and Science and Engineering colleges in Chennai. He has guided several
students to obtain their M.Phil. degree from the University of Madras, Chennai.
He was the most popular and sought-after teacher of Mathematics in Chennai during 1980s for
coaching students for IIT-JEE. He has produced all India 1st rank and several other ranks in IIT-JEE.
He was also a visiting professor at a few leading IIT-JEE training centres in Andhra Pradesh.
Along with his wife C. Vijayakumari, he has written 10 books covering various topics of
Engineering Mathematics catering to the syllabus of Anna University, Chennai, and has also written
“Numerical Analysis”, an all India book, catering to the syllabi of all major universities in India.
Prof. Dr C. Vijayakumari began her career in 1970 as assistant professor
of Mathematics at Government Arts College for Women, Thanjavur, and has
taught at various Government Arts and Science colleges across Tamil Nadu
before retiring as professor of Mathematics from Queen Mary’s College
(Autonomous), Chennai after an illustrious career of spanning 36 years.
As a visiting professor of Mathematics, she has taught the students at
two engineering colleges in Chennai. With a teaching experience spanning
over 40 years, she is an accomplished teacher of Mathematics and Statistics
at both undergraduate and postgraduate levels. She has guided many
students to obtain their M.Phil. degree from the University of Madras,
Chennai and Bharathiar University, Coimbatore.
Along with her husband P. Sivaramakrishna Das, she has co-authored several books on Engineering
Mathematics catering to the syllabus of Anna University, Chennai and has also co-authored “Numerical
Analysis”, an all India book, catering to the syllabi of all major universities in India.
About the Authors

Engineering Mathematics

1.0 INTRODUCTION
The concept of matrices and their basic operations were introduced by the British mathematician
Arthur Cayley in the year 1858. He wondered whether this part of mathematics will ever be used.
However, after 67 years, in 1925, the German physicist Heisenberg used the algebra of matrices in his
revolutionary theory of quantum mechanics. Over the years, the theory of matrices have been found as
an elegant and powerful tool in almost all branches of Science and Engineering like electrical networks,
graph theory, optimisation techniques, system of differential equations, stochastic processes, computer
graphics, etc. Because of the digital computers, usage of matrix methods have become greatly fruitful.
In this chapter, we review some of the basic concepts of matrices. We shall discuss two important
applications of matrices, namely consistency of system of linear equations and the eigen value problems.
1.1 BASIC CONCEPTS
Definition 1.1 Matrix
A rectangular array of mn numbers (real or complex) arranged in m rows (horizontal lines) and
n columns (vertical lines) and enclosed in brackets [ ] is called an m × n matrix.
The numbers in the matrix are called entries or elements of the matrix.
Usually an m × n matrix is written as
A
a a a a
a a a a a
a a a a a
j n
j n
i i i ij in
=
a11 12 13 1 1
21 22 23 2 2
1 2 3
… …
… …
… …
A A A A A
A
A A A
a a a a a
m m m mj mn
1 2 3 … …
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
where aij
is the element lying in the ith
row and jth
column, the first suffix refers to row and the second
suffix refers to column.
The matrix A is briefly written as
A = [aij
]m × n
, i = 1, 2, 3, …, m, j = 1, 2, 3, …, n
If all the entries are real, then the matrix A is called a real matrix.
Definition 1.2 Square Matrix
In a matrix, if the number of rows = number of columns = n, then it is called a square matrix of order n.
If A is a square matrix of order n, then A = [aij
]n × n
, i = 1, 2, 3, …, n; j = 1, 2, 3, …, n.
Definition 1.3 Row Matrix
A matrix with only one row is called a row matrix.
1
Matrices
M01_ENGINEERING_MATHEMATICS-I _CH01_Part A.indd 1 5/30/2016 4:34:37 PM

1.2 ■ Engineering Mathematics
EXAMPLE 1.1
Let A = [a11
a12
a13
… a1n
]. It is a row matrix with n columns. So, it is of type 1 × n.
EXAMPLE 1.2
Let A = [1, 2, 3, 4]. It is a row matrix with 4 columns. So, it is a row matrix of type 1 × 4.
Definition1.4 Column Matrix
A matrix with only one column is called a column matrix.
EXAMPLE 1.3
Let A
a
a
a
an
=
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
11
21
31
1
:
It is a column matrix with n rows. So, it is of type n × 1.
EXAMPLE 1.4
Let A = −
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
1
0
2
1
3
It is a column matrix with 5 rows. So, it is of type 5 × 1.
Definition 1.5 Diagonal Matrix
A square matrix A = [aij
] with all entries aij
= 0 when i ≠ j is is called a diagonal matrix.
In other words a square matrix in which all the off diagonal elements are zero is called a diagonal
matrix.
EXAMPLE 1.5
(1) A
a
ann
=
…
…
…
11 0 0 0
0 0
0 0 0
0 22
a
: : :
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
is a diagonal matrix of order n.
(2) A =
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
2 0 0
0 3 0
0 0 4
is a diagonal matrix of order 3.
(3) A =
−
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
1 0 0 0
0 2 0 0
0 0 3 0
0 0 0 0
is a diagonal matrix of order 4.

Matrices ■ 1.3
Definition 1.6 Scalar Matrix
In a diagonal matrix if all the diagonal elements are equal to a non-zero scalar a, then it is called a
scalar matrix.
EXAMPLE 1.6
A =
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
a
a
a
0 0
0 0
0 0
is a scalar matrix.
Definition 1.7 Unit Matrix or Identity Matrix
In a diagonal matrix, if all the diagonal elements are equal to 1, then it is called a Unit matrix or
identity matrix.
EXAMPLE 1.7
[ ], ,
1
1 0
0 1
1 0 0
0 1 0
0 0 1
⎡
⎣
⎢
⎤
⎦
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
are identity matrices of orders 1, 2, 3 respectively. They are denoted by I1
, I2
, I3
.
In general, In
is the identity matrix of order n.
Definition 1.8 Zero Matrix or Null Matrix
In a matrix (rectangular or square), if all the entries are equal to 0, then it is called a zero matrix or
null matrix.
EXAMPLE 1.8
A B
=
⎡
⎣
⎢
⎤
⎦
⎥ =
⎡
⎣
⎢
⎤
⎦
⎥
0 0
0 0
0 0 0 0
0 0 0 0
, are zero matrices of types 2 × 2 and 2 × 4.
Definition 1.9 Triangular matrix
] is said to be an upper triangular matrix if all the entries below the main
diagonal are zero.
That is aij
= 0 if i j
] is said to be a lower triangular matrix if all the entries above the main
diagonal are zero.
That is aij
= 0 if i j
EXAMPLE 1.9
(1) The matrices A B
=
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
=
−
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
1 2 3
0 1 4
0 0 5
4 1 0 2
0 2 3 1
0 0 0 2
0 0 0 5
and are upper triangular matrices.
(2) The matrices A =
−
⎡
⎣
⎢
⎤
⎦
⎥
2 0
1 0
and B = −
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
1 0 0
2 1 0
0 2 1
are lower triangular matrices.

1.1.1 Basic Operations on Matrices
Definition 1.10 Equality of Matrices
Two matrices A = [aij
] and B = [bij
] of the same type m × n are said to be equal if aij
= bij
for all i, j and
is written as A = B.
Definition 1.11 Addition of Matrices
Let A = [aij
] and B = [bij
] of the same type m × n. Then A + B = [cij
], where cij
= aij
+ bij
for all i and j
and A + B is of type m × n.
EXAMPLE 1.10
If A =
−
⎡
⎣
⎢
⎤
⎦
⎥
1 2 3
0 1 5
and B =
−
⎡
⎣
⎢
⎤
⎦
⎥
1 2 3
1 0 2
, then A B
+ =
− + + +
+ + −
⎡
⎣
⎢
⎤
⎦
⎥ =
⎡
⎣
⎢
⎤
⎦
⎥
1 1 2 2 3 3
0 1 1 0 5 2
0 4 6
1 1 3
We see that A and B are of type 2 × 3 and A + B is also of type 2 × 3.
Definition 1.12 Scalar Multiplication of a Matrix
Let A = [aij
] be an m × n matrix and k be a scalar, then kA = [kaij
].
EXAMPLE 1.11
If A
a a a
a a a
=
⎡
⎣
⎢
⎤
⎦
⎥
11 12 13
21 22 23
, then kA
ka ka ka
ka ka ka
=
⎡
⎣
⎢
⎤
⎦
⎥
11 12 13
21 22 23
.
In particular if k = −1, then − =
− − −
− − −
⎡
⎣
⎢
⎤
⎦
⎥
A
a a a
a a a
11 12 13
21 22 23
.
Multiplication of Matrices
If A and B are two matrices such that the number of columns of A is equal to the number of rows of
B, then the product AB is defined. Two such matrices are said to be conformable for multiplication.
In the product AB, A is known as pre-factor and B is known as post-factor.
Definition 1.13 Let A = [aij
] be an m × p matrix and B = [bij
] be an p × n matrix, then AB is defined and
AB = [cij
] is an m × n matrix, where c a b
ij ik kj
k
p
=
=
∑1
.
That is cij
is the sum of the products of the corresponding elements of the ith
row of A and the jth
column of B.
EXAMPLE 1.12
Let A B
=
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
=
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
1 1 2
0 1 3
2 2 1
1 2
3 1
2 1
and
Since A is of type 3 × 3 and B is of type 3 × 2, AB is defined and AB is of type 3 × 2.
AB =
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
=
⋅ + ⋅ + ⋅ ⋅ + ⋅ +
1 1 2
0 1 3
2 2 1
1 2
3 1
2 1
1 1 1 3 2 2 1 2 1 1 2
2 1
0 1 1 3 3 2 0 2 1 1 3 1
2 1 2 3 1 2 2 2 2 1 1 1
⋅
⋅ + ⋅ + ⋅ ⋅ + ⋅ + ⋅
⋅ + ⋅ + ⋅ ⋅ + ⋅ + ⋅
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
=
=
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
8 5
9 4
10 7

Matrices ■ 1.5
Note If A and B are square matrices of order n, then both AB and BA are deﬁned, but not necessarily
equal. That is, AB ≠ BA, in general.
So, matrix multiplication is not commutative.
1.1.2 Properties of addition, scalar multiplication and multiplication
1. If A, B, C are matrices of the same type, then
(i) A + B = B + A (ii) A + (B + C) = (A + B) + C
(iii) A + 0 = A (iv) A + (−A) = 0
(v) a(A + B) = aA + aB (vi) (a + b)A = aA + bA
(vii) a(bA) = (ab)A for any scalars a, b.
2. If A, B, C are conformable for multiplication, then
(i) a(AB) = (aA)B = A(aB)
(ii) A(BC) = (AB)C
(iii) (A + B)C = AC + BC, where A and B are of type m × p and C is of type p × n.
(iv) If A is a square matrix, then
A2
= A × A, A3
= A2
× A, …, An
= An − 1
× A
Definition 1.14 Transpose of a Matrix
Let A = [aij
] be an m × n matrix. The transpose of A is obtained by interchanging the rows and
columns of A and it is denoted by AT
.
∴ =
A a
T
ji
[ ] is a n × m matrix.
Properties:
(i) (AT
)T
= A (ii) (A + B)T
= AT
+ BT
(iii) (AB)T
= BT
AT
(iv) (aA)T
= aAT
Definition 1.15 Symmetric Matrix
] of order n is said to be symmetric if AT
= A.
This means [aji
] = [aij
] ⇒ aji
= aij
for i, j = 1, 2, …n
Thus, in a symmetric matrix elements equidistant from the main diagonal are the same.
EXAMPLE 1.13
A
a h g
h b f
g f c
=
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
and B =
−
−
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
1 2 3 4
2 0 5 7
3 5 2 8
4 7 8 4
are symmetric matrices of orders 3 and 4.
Definition 1.16 Skew-Symmetric Matrix
] of order n is said to be skew-symmetric if AT
= −A.
This means [aji
] = −[aij
] ⇒ aji
= − aij
for all i, j = 1, 2, …, n
In particular, put j = i, then aii
= − aii
⇒ 2aii
= 0 ⇒ aii
= 0 for all i = 1, 2, …, n
So, in a skew-symmetric matrix, the diagonal elements are all zero and elements equidistant from
the main diagonal are equal in magnitude, but opposite in sign.

EXAMPLE 1.14
A B
=
−
⎡
⎣
⎢
⎤
⎦
⎥ =
−
−
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
0 1
1 0
0 2 3
2 0 4
3 4 0
and are skew-symmetric matrices of orders 2 and 3.
Definition 1.17 Non-Singular Matrix
A Square matrix A is said to be non-singular if A ≠ 0 ( A means determinant of A).
If A = 0, then A is singular.
Definition 1.18 Minor and Cofactor of an Element
Let A = [aij
] be a square matrix of order n. If we delete the row and column of the element aij
, we get
a square submatrix of order (n − 1).
The determinant of this submatrix is called the minor of the element aij
and is denoted by Mij
.
The cofactor of aij
in A is A M
ij
i j
ij
= − +
( )
1
EXAMPLE 1.15
A = −
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
1 6 2
0 2 4
3 1 2
The cofactor of a11
= 1 is A11
1 1
1
2 4
1 2
= −
−
+
( ) = −4 −4 = −8
The cofactor of a12
= 6 is A12
1 2
1
0 4
3 2
= − +
( ) = − (−12) = 12
The cofactor of a32
= 1 is A32
3 2
1
1 2
0 4
= − +
( ) = − (4 −0) = −4
Similarly, we can determine the cofactors of other elements.
Definition 1.19 Adjoint of a Matrix
Let A = [aij
] be a square matrix. The adjoint of A is defined as the transpose of the matrix of cofactors
of the elements of A and it is denoted by adj A.
Thus, adj A
A A A
A A A
A A A
n
n
n n nn
T
=
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
11 12 1
21 21 2
1 2
…
…
…
: : :
Properties: If A and B are square matrices of order n, then
(i) adj AT
= (adj A)T
(ii) (adj A) A = A (adj A) = A In
.
(iii) adj(AB) = (adj A) (adj B)
Using property (ii), we define inverse.
Definition 1.20 Inverse of a Matrix
If A is a non-singular matrix, then the inverse of A is defined as
adj A
A
and it is denoted by A−1
.

Matrices ■ 1.7
∴ A
A
A
−
=
1 adj
EXAMPLE 1.16
Find the inverse of A = −
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
1 6 2
0 2 4
3 1 2
.
Solution.
Given A = −
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
1 6 2
0 2 4
3 1 2
∴ A = −
1 6 2
0 2 4
3 1 2
= 1(−4 −4) −6(0 − 12) + 2(0 + 6) = −8 + 72 + 12 = 76 ≠ 0
Since A ≠ 0, A is non-singular and hence A−1
exists and A
A
A
−
=
1 adj
.
We shall ﬁnd the cofactors of the elements of A
A A
A
11
1 1
12
1 2
13
1
2 4
1 2
4 4 8 1
0 4
3 2
0 12 12
= −
−
= − − = − = − = − − =
=
+ +
( ) ( ) , ( ) ( )
(−
−
−
= + = = − = − − = −
= −
+ +
+
1
0 2
3 1
0 6 6 1
6 2
1 2
12 2 10
1
1 3
21
2 1
22
2
) ( ) , ( ) ( )
( )
A
A 2
2
23
2 3
31
3 1
1 2
3 2
2 6 4 1
1 6
3 1
1 18 17
1
6 2
2 4
= − = − = − = − − =
= −
−
+
+
( ) , ( ) ( )
( )
A
A =
= + = = − = − − = −
= −
−
= − −
+
+
( ) , ( ) ( )
( ) (
24 4 28 1
1 2
0 4
4 0 4
1
1 6
0 2
2
32
3 2
33
3 3
A
A 0
0 2
) = −
∴
8 12 6
10 4 17
28 4 2
8 10 28
12 4 4
6 17
=
−
− −
− −
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
=
− −
− −
−
adj A
T
2
2
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
∴ A −
=
− −
− −
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
1 1
76
8 10 28
12 4 4
6 17 2
1.2 COMPLEX MATRICES
A matrix with at least one element as complex number is called a complex matrix.
Let A = [aij
] be a complex matrix.
The conjugate matrix of A is denoted by A and A aij
= ⎡
⎣ ⎤
⎦.

EXAMPLE 1.17
A
i i
i
=
−
−
⎡
⎣
⎢
⎤
⎦
⎥
2 2
3 2 0 3
is a complex matrix.
The conjugate of A is A
i i
i
i i
i
=
−
−
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
=
−
+
⎡
⎣
⎢
⎤
⎦
⎥
2 2
3 2 0 3
2 2
3 2 0 3
[{ conjugate of a + ib = a − ib]
We denote A
T
( ) by A*.
∴ A* is the transpose of the conjugate of A.
In the above example
A
i i
i
*
=
− +
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
2 3 2
0
2 3
Note We have A A A A
T T T
⎡
⎣ ⎤
⎦ = ⎡
⎣ ⎤
⎦ ∴ = ⎡
⎣ ⎤
⎦
∗
∴ If A a A a A a
ji
T
ji
T
ji
= = ⎡
⎣ ⎤
⎦
[ ], [ ],
then =[ ] ∴ A ∗
= [ ]
aji
Definition 1.21 Hermitian Matrix
A complex square matrix A is said to be a Hermitian matrix if A* = A and Skew-Hermitian matrix if
A* = −A.
A Hermitian matix is also denoted by AH
.
If A = [aij
], then A aji
* [ ]
= ∴ A* = A ⇒ aji = aij
for all i and j
Put j = i, then aii = aii
⇒ aii
are real numbers.
So, the diagonal elements of a Hermitian matrix are real numbers.
The elements equidistant from the main diagonal are conjugates.
A* = −A ⇒ aji = −aij
for all i and j
Put j = i, then aii = −aii
If aii
= a + ib, then aii = a − ib
∴ a − ib = −(a + ib) ⇒ 2a = 0 ⇒ a = 0
∴ aii
= ib, which is purely imaginary if b ≠ 0 and 0 if b = 0.
∴ the diagonal elements of a Skew-Hermitian matrix are all purely imaginary or 0 and the elements
equidistant from the main diagonal are conjugates with opposite sign.
Properties: If A and B are complex matrices, then
1. A A
( ) = , 2. A B A B
+ = + 3. a a
A A
=
4. AB A B
= 5. (A*).* = A 6. (A + B).* = A* + B*
7. (aA).* = aA * 8. (AB).* = B*A*
Definition 1.22 Unitary Matrix
A complex square matrix is said to be unitary if AA* = A*A = I
From the deﬁnition it is obvious that A* is the inverse of A.
∴ A* = A−1

Matrices ■ 1.9
EXAMPLE 1.18
Show that A
i
i
5
2 2
1
1 3
3 2
⎡
⎣
⎢
⎤
⎦
⎥ is a Hermitian matrix.
Solution.
Given A =
− −
+
⎡
⎣
⎢
⎤
⎦
⎥
1 3
3 2
i
i
∴ A A
i
i
T
T
* = ( ) =
− −
+
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
1 3
3 2
=
− +
−
⎡
⎣
⎢
⎤
⎦
⎥ =
− −
+
⎡
⎣
⎢
⎤
⎦
⎥
1 3
3 2
1 3
3 2
i
i
i
i
T
= A
∴ A is Hermitian matrix.
EXAMPLE 1.19
Show that B
i i
i i
i i
5
2 1
1 2
2 1 2
1 3 3 2
3 0 2 3
3 2 2 3 2
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
is a Hermitian matrix.
Solution.
Since the diagonal elements are real and elements equidistant from the main diagonal are conjugates,
B is a Hermitian matrix.
EXAMPLE 1.20
Show that A
i i
i
5
1
2 2
2 1
1 0
( )
⎡
⎣
⎢
⎤
⎦
⎥ is a Skew-Hermitian matrix.
Solution.
Given A
i i
i
=
+
− −
⎡
⎣
⎢
⎤
⎦
⎥
2 1
1 0
( )
Since the diagonal elements are purely imaginary or zero and (1 + i) and − (1 − i) are conjugates with
opposite sign, A is Skew-Hermitian matrix.
EXAMPLE 1.21
Show that B
i i i
i i
i i i
5
1 2
2 2 1
2 1 2 2
2 1 2 5
1 0 2 3
2 5 2 3 3
( )
( ) ( )
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
is skew-Hermitian.
Solution.
Given B
i i i
i i
i i i
=
+ −
− − +
− + − −
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
2 1 2 5
1 0 2 3
2 5 2 3 3
( )
( ) ( )
In B, the diagonal elements are purely imaginary or zero and the elements equidistant from the main
diagonal are conjugates with opposite sign. So, B is skew-Hermitian matrix.
Note If A is a real matrix, then the deﬁnition of unitary
⇒ AAT
= AT
A = I.
In this case A is called an orthogonal matrix. So, if A is an orthogonal matrix, then AT
= A−1
.

WORKED EXAMPLES
EXAMPLE 1
If A
i i
i i
5
1 2 1
2 2
2 3 1 3
5 4 2
⎡
⎣
⎢
⎤
⎦
⎥ , then show that AA* is a Hermitian matrix.
Solution.
Given A
i i
i i
=
+ − +
− −
⎡
⎣
⎢
⎤
⎦
⎥
2 3 1 3
5 4 2
∴ A* = A
i i
i i
T
T
[ ] =
− − −
− − +
⎡
⎣
⎢
⎤
⎦
⎥
2 3 1 3
5 4 2
=
− −
−
− − +
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
2 5
3
1 3 4 2
i
i
i i
We have to prove AA* is a Hermitian matrix.
That is to prove (AA*)* = AA*
Now AA
i i
i i
i
i
i i
* =
+ − +
− −
⎡
⎣
⎢
⎤
⎦
⎥
− −
−
− − +
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
2 3 1 3
5 4 2
2 5
3
1 3 4 2
=
+ − + ⋅ + − + − − + − + − + − + +
( )( ) ( )( ) ( )( ) ( ) ( ) (
2 2 3 3 1 3 1 3 2 5 3 1 3 4 2
i i i i i i i i
i
i i i i i i i i
)
( ) ( )( ) ( )( ) ( ) ( )( )
− − + ⋅ + − − − − − + − + − +
⎡
⎣
⎢ 5 2 3 4 2 1 3 5 5 4 2 4 2
⎤
⎤
⎦
⎥
=
+ + + + − − − − + +
− + + − − + − +
2 1 9 1 3 10 5 3 4 10 6
10 5 3 4 10 6 25 4
2 2 2
2 2 2
i i i i
i i i i i +
+
⎡
⎣
⎢
⎤
⎦
⎥
=
− + −
− − −
⎡
⎣
⎢
⎤
⎦
⎥ =
− +
− −
⎡
⎣
⎢
⎤
2
24 14 2 6
14 2 6 46
24 20 2
20 2 46
2
i
i
i
i ⎦
⎦
⎥ = −
[ ]
{ i2
1
∴ ( *)*
AA
i
i
T
=
− +
− −
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
24 20 2
20 2 46
=
− −
− +
⎡
⎣
⎢
⎤
⎦
⎥ =
− +
− −
⎡
⎣
⎢
⎤
⎦
⎥
24 20 2
20 2 46
24 20 2
20 2 46
i
i
i
i
T
= AA*
⇒ (AA*)* = AA*
Hence, AA* is a Hermitian matrix.
EXAMPLE 2
Show that every square complex matrix can be expressed uniquely as P + iQ, where P and Q are
Hermitian matrices.
Solution.
Let A be any square complex matrix.
We shall rewrite A as
A A A i
i
A A
= + + −
⎡
⎣
⎢
⎤
⎦
⎥
1
2
1
2
[ *] ( *)

Matrices ■ 1.11
Put P A A Q
i
A A
= + = −
1
2
1
2
( *), ( *), then A = P + iQ.
We shall now prove P and Q are Hermitian.
Now, P A A A A A A P
*
( *) ( * ( *)* ( * )
*
= +
⎡
⎣
⎢
⎤
⎦
⎥ = +
⎡
⎣
⎢
⎤
⎦
⎥ = + =
1
2
1
2
1
2
∴ P is Hermitian.
and Q A A
i
A A
i
A A
i
A A Q
* ( *) ( * ( *)* [ * ] ( *)
*
= −
⎡
⎣
⎢
⎤
⎦
⎥ = −
[ ]= − − = − =
1
2
1
2
1
2
1
2
i
∴ Q is Hermitian.
We shall now prove the uniqueness of the expression A = P + iQ.
If possible, let A = R + iS (1)
where R and S are Hermitian matrices.
∴ R* = R and S* = S
Now, A* = (R + iS)* = R* + (iS)* = R* − iS* = R − iS [by property] (2)
(1) + (2) ⇒ A + A* = 2R ⇒ R =
1
2
( *)
A A P
+ =
(1) − (2) ⇒ A − A* = 2iS ⇒ S =
1
2i
A A Q
( *)
− =
∴ the expression A = P + iQ is unique.
EXAMPLE 3
If A is any square complex matrix, prove that (1) A 1 A* is Hermitian and (ii) A 5 B 1 C, where
B is Hermitian and C is Skew-Hermitian.
Solution.
Given A is a square complex matrix.
(i) Let P = A + A*
∴ P* = (A + A*)* = A* + (A*)* = A* + A = A + A* = P [by property]
∴ P is Hermitian
Hence, A + A* is Hermitian.
To prove (ii): Since A is square complex matrix, we can write A as
A A A A A
= + + −
1
2
1
2
( *) ( *) = B + C
where B A A
= +
1
2
( *) is Hermitian by part (i) and C A A
= −
1
2
( *)
⇒ C A A A A
* ( *) ( * ( *)*
*
= −
⎡
⎣
⎢
⎤
⎦
⎥ = −
[ ]
1
2
1
2
= − = − − = −
1
2
1
2
[ * ] [ *]
A A A A C
∴ C is Skew- Hermitian.

EXAMPLE 4
If A 5
1
2 1
0 1 2
2 0
i
i
1
⎡
⎣
⎢
⎤
⎦
⎥ , then show that (I 2 A) (I 1 A)21
is a unitary matrix.
Solution.
Given A
i
i
=
+
− +
⎡
⎣
⎢
⎤
⎦
⎥
0 1 2
1 2 0
=
+
− −
⎡
⎣
⎢
⎤
⎦
⎥
0 1 2
1 2 0
i
i
( )
. Let I =
⎡
⎣
⎢
⎤
⎦
⎥
1 0
0 1
.
∴ I A
i
i
+ =
⎡
⎣
⎢
⎤
⎦
⎥ +
+
− −
⎡
⎣
⎢
⎤
⎦
⎥
1 0
0 1
0 1 2
1 2 0
( )
=
+
− −
⎡
⎣
⎢
⎤
⎦
⎥
1 1 2
1 2 1
i
i
( )
I A
i
i
i i
+ =
+
− −
= + − + = + + = ≠
1 1 2
1 2 1
1 1 2 1 2 1 1 4 6 0
( )
( )( )
∴ Inverse of I + A exists and ( )
( )
I A
I A
I A
+ =
+
+
−1 adj
adj( )
( )
( )
I A
i
i
i
i
T
+ =
−
− +
⎡
⎣
⎢
⎤
⎦
⎥ =
− +
−
⎡
⎣
⎢
⎤
⎦
⎥
1 1 2
1 2 1
1 1 2
1 2 1
∴ ( )
( )
I A
i
i
+ =
− +
−
⎡
⎣
⎢
⎤
⎦
⎥
−1 1
6
1 1 2
1 2 1
∴ I A
i
i
− =
⎡
⎣
⎢
⎤
⎦
⎥ −
+
− +
⎡
⎣
⎢
1 0
0 1
0 1 2
1 2 0
⎤
⎤
⎦
⎥ =
− +
−
⎡
⎣
⎢
⎤
⎦
⎥
1 1 2
1 2 1
( )
i
i
∴ (I A I A
i
i
i
i
− + =
− +
−
⎡
⎣
⎢
⎤
⎦
⎥
− +
−
⎡
⎣
⎢
⎤
⎦
⎥
=
−
−
)( )
( ) ( )
1 1
6
1 1 2
1 2 1
1 1 2
1 2 1
1
6
1 (
( )( ) ( ) ( )
( ) ( ) ( )( )
1 2 1 2 1 2 1 2
1 2 1 2 1 2 1 2 1
+ − − + − +
− + − − − + +
⎡
⎣
⎢
i i i i
i i i i
⎤
⎤
⎦
⎥
=
− + − +
− − + +
⎡
⎣
⎢
⎤
⎦
⎥ =
− − +
1
6
1 1 4 2 1 2
2 1 2 1 4 1
1
6
4 2 1 2
2
( ) ( )
( ) ( )
( )
(
i
i
i
1
1 2 4
− −
⎡
⎣
⎢
⎤
⎦
⎥ =
i
B
)
, say
Now, B
i
i
T
*
( )
( )
=
− − +
− −
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
1
6
4 2 1 2
2 1 2 4
=
− − −
+ −
⎡
⎣
⎢
⎤
⎦
⎥ =
− +
− − −
⎡
⎣
⎢
⎤
⎦
⎥
1
6
4 2 1 2
2 1 2 4
1
6
4 2 1 2
2 1 2 4
( )
( )
( )
( )
i
i
i
i
T
To prove B = (I − A) (I + A)−1
is unitary, verify BB* = I
Now, BB
i
i
i
i
*
( )
( )
( )
( )
=
− − +
− −
⎡
⎣
⎢
⎤
⎦
⎥
− +
− − −
⎡
⎣
⎢
⎤
⎦
⎥
1
36
4 2 1 2
2 1 2 4
4 2 1 2
2 1 2 4
=
+ + − − + + +
− − + − +
1
36
16 4 1 2 1 2 8 1 2 8 1 2
8 1 2 8 1 2 4 1 2
( )( ) ( ) ( )
( ) ( ) (
i i i i
i i i)
)( )
1 2 16
− +
⎡
⎣
⎢
⎤
⎦
⎥
i

Matrices ■ 1.13
( )
( )
1
36
16 4 1 4 0
0 4 1 4 6
1
36
36 0
0 36
=
+ +
+ +
⎡
⎣
⎢
⎤
⎦
⎥ =
⎡
⎣
⎢
⎢
⎤
⎦
⎥ =
⎡
⎣
⎢
⎤
⎦
⎥ =
1 0
0 1
I.
∴ B is unitary.
Hence, (I − A)(I + A)−1
is unitary.
Note Another method: To prove B is unitary, verify B* = B−1
EXERCISE 1.1
1. If A B A B
+ =
⎡
⎣
⎢
⎤
⎦
⎥ − =
⎡
⎣
⎢
⎤
⎦
⎥
7 0
2 5
3 0
0 3
, , find A and B
2. Find x, y, z and w if
3
6
1 2
4
3
x y
z w
x
w
x y
z w
⎡
⎣
⎢
⎤
⎦
⎥ =
−
⎡
⎣
⎢
⎤
⎦
⎥ +
+
+
⎡
⎣
⎢
⎤
⎦
⎥
3. If matrix A has x rows and x +5 columns and B has y rows and 11 − y columns such that both AB and BA
exist, then find x and y.
4. If A =
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
2 3 4
1 2 3
1 1 2
and B = −
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
1 3 0
1 2 1
0 0 2
, then find AB and BA and test their equality.
5. If A =
−
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
0
2
2
0
tan
tan
a
a
, show that I A I A
+ = −
−
⎡
⎣
⎢
⎤
⎦
⎥
[ ]
cos sin
sin cos
a a
a a
6. If A =
−
⎡
⎣
⎢
⎤
⎦
⎥
cos sin
sin cos
a a
a a
, then verify that AAT
= I2
.
7. If A is a square matrix, then show that A can be expressed as A = P + Q, where P is symmetric and Q is
skew-symmetric.
Hint: Take P
A A
Q
A A
T T
=
+
=
−
⎡
⎣
⎢
⎤
⎦
⎥
2 2
,
8. If A =
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
2 0 1
2 1 3
1 1 0
and f(x) = x2
− 5x + 6, then find f(A).
9. If A =
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
1 1 1
2 3 0
18 2 10
, then prove that A(adj A) =
0 0 0
0 0 0
0 0 0
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
.
10. Find the inverse of A =
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
1 0 0
3 3 0
5 2 1
in terms of adj A.
11. Show that A
i i
i i
=
+ − +
+ −
⎡
⎣
⎢
⎤
⎦
⎥
1
2
1 1
1 1
is unitary.
12. If A and B are orthogonal matrices of the same order, prove that AB is orthogonal.
[Hint: AAT
= I, BBT
= I. Compute AB(AB)T
= A(BBT
)AT
= AAT
= I].

13. If A and B are Hermitian matrices of the same order, prove that
(i) A + B is Hermitian (ii) AB + BA is Hermitian
(iii) iA is Skew-Hermitian (iv) AB − BA is Skew-Hermitian
14. Find the inverse of the following matrices.
(i)
2 1 4
3 0 1
1 1 2
−
−
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
(ii)
4 3 3
1 0 1
4 4 3
− −
− − −
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
15. If A =
3 3 4
2 3 4
0 1 1
−
−
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
, then show that A3
= A−1
.
ANSWERS TO EXERCISE 1.1
1. A B
=
⎡
⎣
⎢
⎤
⎦
⎥ =
⎡
⎣
⎢
⎤
⎦
⎥
5 0
1 4
2 0
1 1
, 2. x = 2, y = 4, z = 1, w = 3 3. x = 3, y = 8
4. AB ≠ BA 8. f A
( ) =
− −
− − −
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
1 1 3
1 1 10
5 4 4
10. A−
= −
−
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
1
1 0 0
1
1
3
0
3
2
3
1
14. (i) A−
=
−
−
− − −
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
1
1 6 1
5 8 14
3 1 3
(ii) A−
= − −
− −
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
1
4 3 3
1 0 1
4 4 3
1.3 RANK OF A MATRIX
Let A = [aij
] be an m × n matrix. A matrix obtained by omitting some rows and columns of A is called
a submatrix of A.
The determinant of a square submatrix of order r is called a minor of order r of A.
EXAMPLE 1.22
Consider A =
−
− −
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
2 3 4 1
0 3 4 0
3 2 1 2
Omitting the fourth column, we get the submatrix A1
2 3 4
0 3 4
3 2 1
=
− −
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
and A1
2 3 4
0 3 4
3 2 1
=
− −
is a
minor of order 3.
Omitting the ﬁrst and third columns and the third row, we get the submatrix A2
3 1
3 0
=
−
⎡
⎣
⎢
⎤
⎦
⎥ and
A2
3 1
3 0
=
−
is a minor of order 2. Since A2 = 3 ≠ 0, it is called a non-vanishing minor of order 2.
But
3 4
3 4
0
= , so it is called a vanishing minor of order 2.

Matrices ■ 1.15
Definition 1.23 Rank of a Matrix
Let A be an m × n matrix. A is said to be of rank r if (1) at least one minor of A of order r is not zero
and (ii) every minor of A of order (r + 1) (and higher order) is zero.
The rank of A is denoted by r(A) or r(A).
Note
(1) The definition says rank of A is the order of the largest non-vanishing minor of A.
(2) The Rank of Zero matrix is zero.
(3) All non-zero matrices have rank ≥ 1.
(4) The rank of an m × n matrix is less than or equal to the min {m, n}.
(5) r(A) = r(AT
)
(6) If In
is the unit matrix of order n, then In = ≠
1 0 and so, r(In
) = n.
To find the rank of a matrix A, we have to identify the largest non-vanishing minor. This process
involves a lot of computations and so it is tedious for matrices of large type.
To reduce the computations, we apply elementary transformations and transform the given matrix
to a convenient form, namely Echelon form or normal form.
Elementary transformations
1. Interchange of any two rows (or columns)
2. Multiplication of elements of any row (or column) by a non-zero number k.
3. Addition to the elements of a row (column), the corresponding elements of another row (column)
multiplied by a fixed number.
Note When an elementary transformation is applied to a row, it is called a row transformation and
when it is applied to a column, it is called a column transformation.
Notation: The following symbols will be used to denote the elementary row operations.
(i) Ri
↔ Rj
means ith
row and jth
row are interchanged.
(ii) Ri
→ kRi
means the elements of ith
row is multiplied by k (≠0)
(iii) Ri
→ Ri
+ kRj
means the jth
row is multiplied by k and added to the ith
row.
Similarly we indicate the column transformations by Ci
↔ Cj
, Ci
→ kCi
, Ci
→ Ci
+ kCj
Definition 1.24 Equivalent Matrices
Two matrices A and B of the same type are said to be equivalent if one matrix can be obtained from the
other by a sequence of elementary row (column) transformations. Then we write A ~ B.
Results:
1. The Rank of a matrix is unaffected by elementary transformations.
2. Equivalent matrices have the same rank.
Definition 1.25 Echelon Matrix
A matrix is called a row-echelon matrix if (1) all zero rows (i.e., rows with zero elements only), if any,
are on the bottom of the matrix and (ii) each leading non-zero element is to the right of the leading
non-zero element in the preceding row.

EXAMPLE 1.23
A =
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
1 2 3
0 1 2
0 0 0
, B =
−
−
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
2 1 1 0
0 0 1 2
0 0 0 0
0 0 0 0
,
C =
−
−
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
1 1 0 4 5
0 1 2 1 3
0 0 0 6 1
0 0 0 0 0
and D =
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
1 2 3
0 2 1
0 0 2
are row echelon matrices.
Note
Triangular matrix is a special case of an echelon matrix.
Result: If a matrix A is equivalent to a row echelon matrix B, then r(A) = the number of non-zero rows
of B.
In the above examples, r(A) = 2, r(B) = 2, r(C) = 3, r(D) = 3.
WORKED EXAMPLES
EXAMPLE 1
Find the rank of the matrix A 5
1 2 3 0
2 4 3 2
3 2 1 3
6 8 7 5
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
, by reducing to an echelon matrix.
Solution.
Given A =
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
1 2 3 0
2 4 3 2
3 2 1 3
6 8 7 5
∼
1 2 3 0
0 0 3 2
0 4 8 3
0 4 11 5
2
3
2 2 1
3 3
−
− −
− −
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
→ + −
→ + −
R R R
R R R
( )
( ) 1
1
4 4 1
4 4 3
6
1 2 3 0
0 0 3 2
0 4 8 3
0 0 3 2
1 2
R R R
R R R
→ + −
−
− −
−
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
→ −
( )
∼
∼
3
3 0
0 4 8 3
0 0 3 2
0 0 3 2
2 3
− −
−
−
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
↔
R R
∼
1 2 3 0
0 4 8 3
0 0 3 2
0 0 0 0 4 4 3
− −
−
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
→ −
R R R
= B, which is a row echelon matrix.
∴ r(A) = the number of non-zero rows in B = 3

Matrices ■ 1.17
EXAMPLE 2
Determine the rank of the matrix A 5
2
2
2
1 2 3 1
3 6 9 3
2 4 6 2
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
Solution.
Given A =
−
−
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
1 2 3 1
3 6 9 3
2 4 6 2
∼
1 2 3 1
0 0 0 0
0 0 0 0
3
2
2 2 1
3 3 1
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
→ + −
→ + −
R R R
R R R
( )
( )
= B, which is a row echelon matrix.
∴ r(A) = the number of non-zero rows in B = 1
EXAMPLE 3
Find the value of k if the rank of the matrix
6 3 5 9
5 2 3 6
3 1 2 3
2 1 1 k
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
is 3.
Solution.
Let A
k
=
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
6 3 5 9
5 2 3 6
3 1 2 3
2 1 1
∼
∼
1
1
2
5
6
3
2
5 2 3 6
3 1 2 3
2 1 1
1
6
1
1
2
5
6
3
2
0
1
2
7
6
1 1
k
R R
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
→
− − −
−
− − −
− −
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
→ + −
3
2
0
1
2
3
6
3
2
0 0
4
6
3
5
2 2 1
k
R R R
( )
R
R R R
R R R
3 3 1
4 4 1
3
2
→ + −
→ + −
( )
( )
∼
1
1
2
5
6
3
2
0
1
2
7
6
3
2
0 0
4
6
0
0 0
4
6
3
3
− − −
− −
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
→
k
R R
R R
3 2
−

k R4
1
1
2
5
6
3
2
0
1
2
7
6
3
2
0 0
4
6
0
0 0 0 3
− − −
−
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥ →
∼
R
R R
4 3
+
= B
Given r(A) = 3. So, the number of non-zero rows of B should be 3.
∴ k − 3 = 0 ⇒ k = 3
Definition 1.26 Elementary Matrix
A matrix obtained from a unit matrix by performing a single elementary row (column) transformation
is called an elementary matrix.
Since unit matrices are non-singular square matrices, elementary matrices are also non-singular.
EXAMPLE 1.24
I3
1 0 0
0 1 0
0 0 1
=
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
z
1 0 0
0 1 1
0 0 1 3 3 2
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥ → +
C C C
This is an elementary matrix.
Similarly,
1 0 0
0 3 0
0 0 1
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
got by R2
→ 3R2
is an elementary matrix.
Definition 1.27 Normal form of a Matrix
Any non-zero matrix A of rank r can be reduced by a sequence of elementary transformations to the
form
Ir 0
0 0
⎡
⎣
⎢
⎤
⎦
⎥, where Ir
is a unit matrix of order r.
This form is called a normal form of A.
Other normal forms are Ir
,
Ir
0
⎡
⎣
⎢
⎤
⎦
⎥, [Ir
, 0].
Theorem 1.1
Let A be an m × n matrix of rank r. Then there exist non-singular matrices P and Q of orders m and n
respectively such that PAQ =
Ir 0
0 0
⎡
⎣
⎢
⎤
⎦
⎥
Note Each elementary row transformation of A is equivalent to pre multiplying A by the corresponding
elementary matrix. Each elementary column transformation is equivalent to post multiplying A
by the corresponding elementary matrix. So, there exists elementary matrices P1
, P2
, …, Pk
and
Q1
, Q2
, …, Qt
such that
P1
P2
… Pk
A Q1
Q2
… Qt
=
Ir 0
0 0
⎡
⎣
⎢
⎤
⎦
⎥

Matrices ■ 1.19
⇒ PAQ =
Ir 0
0 0
⎡
⎣
⎢
⎤
⎦
⎥
where P = P1
P2
… Pk
, Q = Q1
Q2
… Qt
Working rule to ﬁnd normal form and P, Q:
Let A be a non-zero m × n matrix write A = Im
AIn
(which is obviously true). Reduce A on the L. H. S to
normal form by applying elementary row and column transformations on A.
Each elementary row transformation of A will be applied to Im
on R. H. S and each elementary
column transformation of A will be applied to In
on R. H. S.
After a sequence of suitable applications of elementary transformations, we get
Ir 0
0 0
⎡
⎣
⎢
⎤
⎦
⎥ = PAQ.
Then the rank of A is the rank of Ir
= r
WORKED EXAMPLES
EXAMPLE 1
Reduce the matrix
0 1 2 1
1 2 3 1
3 1 1 3
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
to normal form and hence ﬁnd the rank.
Solution.
Let A =
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
0 1 2 1
1 2 3 2
3 1 1 3
∼
∼
1 0 2 1
2 1 3 2
1 3 1 3
1 0 0 0
2 1 1 0
1 3 1 2
1 2
3 3
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
↔
−
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
→
C C
C C +
+ −
→ −
( )
2 1
4 4 1
C
C C C
∼
∼
1 0 0 0
0 1 1 0
0 3 1 2
2
1 0 0 0
0 1 0 0
0 3 2 2
2 2 1
3 3 1
−
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
→ + −
→ −
R R R
R R R
( )
⎡
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥ → +
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥ → + −
C C C
R R R
3 3 2
3 3 2
1 0 0 0
0 1 0 0
0 0 2 2 3
1 0 0
∼
∼
( )
0
0
0 1 0 0
0 0 1 1
1
2
3 3
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥ →
R R

1 0 0 0
0 1 0 0
0 0 1 0 4 4 3
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥ → −
C C C
∼
=
= [ : ]
I3 0
This is the normal form of A and so the r(A) = 3
EXAMPLE 2
Let A 5
2 2
1 1 1
1 1 1
3 1 1
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
.Find matrices P and Q such that PAQ is in the normal form.Also ﬁnd rank
of A.
Solution.
Given A =
− −
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥ ×
1 1 1
1 1 1
3 1 1 3 3
Consider A = I3
AI3
⇒
1 1 1
1 1 1
3 1 1
1 0 0
0 1 0
0 0 1
1 0 0
0 1 0
0 0 1
− −
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
=
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
A ⎥
⎥
⎥
⎥
Our aim is to reduce the LHS matrix to normal form.
Also row operations to be applied to pre factor and column operations to be applied to post factor.
Apply,
and to post factor
C C C
C C C
2 2 1
3 3 1
1 0 0
1 2 2
3 4 4
→ +
→ +
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
=
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
1 0 0
0 1 0
0 0 1
1 1 1
0 1 0
0 0 1
A
R R R
R R R
2 2 1
3 3 1
3
1 0 0
0 2 2
0 4 4
1
→ −
→ + −
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
=
( )
and to pre factor
0
0 0
1 1 0
3 0 1
1 1 1
0 1 0
0 0 1
−
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
A
R R R
3 3 2
2
1 0 0
0 2 2
0 0 0
1 0 0
1 1 0
1
→ + −
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
= −
−
( )
and to pre factor −
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
2 1
1 1 1
0 1 0
0 0 1
A
R R
2 2
1
2
1 0 0
0 1 1
0 0 0
1 0 0
1
2
1
2
0
1 2 1
→
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
= −
− −
and to pre factor
⎡
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
A
1 1 1
0 1 0
0 0 1

Matrices ■ 1.21
C C C
3 3 2
1 0 0
0 1 0
0 0 0
1 0 0
1
2
1
2
0
1
→ −
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
= −
− −
and to post factor
2
2 1
1 1 0
0 1 1
0 0 1
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
A
⇒
I
PAQ
2 0
0 0
⎡
⎣
⎢
⎤
⎦
⎥ =
This shows r(A) = 2, and
P Q
= −
− −
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
= −
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
1 0 0
1
2
1
2
0
1 2 1
1 1 0
0 1 1
0 0 1
,
EXAMPLE 3
Let A 5
2 2
2
2
1 1 1 2
4 2 2 1
2 2 0 2
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
. Find the non-singular matrices P and Q, such that PAQ is in the
normal form.Also ﬁnd the rank of A.
Solution.
Given A =
− −
−
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥ ×
1 1 1 2
4 2 2 1
2 2 0 2 3 4
Consider A = I3
AI4
⇒
1 1 1 2
4 2 2 1
2 2 0 2
1 0 0
0 1 0
0 0 1
1 0 0 0
0 1 0 0
0 0 1
− −
−
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
=
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
A
0
0
0 0 0 1
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
Our aim is to reduce the LHS matrix to normal form. Also row operations to be applied to pre factor
and column operations to be applied to post factor.
C C C
C C C
C C C
2 2 1
3 3 1
4 4 1
2
1 0 0 0
4 6 6 9
2 4 2
→ +
→ +
→ + −
−
( )
and to post factor
−
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
=
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
−
⎡
⎣
⎢
⎢
⎢
⎢
⎤
6
1 0 0
0 1 0
0 0 1
1 1 1 2
0 1 0 0
0 0 1 0
0 0 0 1
A
⎦
⎦
⎥
⎥
⎥
⎥
R R R
R R R
2 2 1
3 3 1
4
2
1 0 0 0
0 6 6 9
0 4 2 6
→ + −
→ + − −
−
⎡
⎣
⎢
( )
( )
and to pre factor
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
= −
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
−
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
1 0 0
4 1 0
2 0 1
1 1 1 2
0 1 0 0
0 0 1 0
0 0 0 1
A
⎥
⎥
⎥

R R
R R
2 2
3 3
1
3
1
2
1 0 0 0
0 2 2 3
0 2 1 3
1 0
→
→
−
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
=
and to pre factor
0
0
4
3
1
3
0
1 0
1
2
1 1 1 2
0 1 0 0
0 0 1 0
0 0 0 1
−
−
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
−
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
A
⎥
⎥
C C C
4 4 2
3
2
1 0 0 0
0 2 2 0
0 2 1 0
1 0 0
4
3
1
→ +
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
= −
and to post factor
3
3
0
1 0
1
2
1 1 1
1
2
0 1 0
3
2
0 0 1 0
0 0 0 1
−
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
−
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
A ⎥
⎥
⎥
⎥
⎥
C C C
3 3 2
1 0 0 0
0 2 0 0
0 2 1 0
1 0 0
4
3
1
3
→ −
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
= −
and to post factor
0
0
1 0
1
2
1 1 0
1
2
0 1 1
3
2
0 0 1 0
0 0 0 1
−
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
−
−
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
A ⎥
⎥
⎥
⎥
⎥
R R R
3 3 2
1 0 0 0
0 2 0 0
0 0 1 0
1 0 0
4
3
1
3
0
→ −
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
= −
and to pre factor
1
1
3
1
3
1
2
1 1 0
1
2
0 1 1
3
2
0 0 1 0
0 0 0 1
−
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
−
−
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
A
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
R R
R R
2 2
3 3
1
2
1
1 0 0 0
0 1 0 0
0 0 1 0
1 0
→
→ −
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
=
( )
and to pre factor
0
0
2
3
1
6
0
1
3
1
3
1
2
1 1 0
1
2
0 1 1
3
2
0 0 1 0
0 0 0 1
−
− −
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
−
−
⎡
⎣
⎢
A
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⇒ [I3
: 0] = PAQ,
where P Q
= −
− −
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
=
−
−
1 0 0
2
3
1
6
0
1
3
1
3
1
2
1 1 0
1
2
0 1 1
3
2
0 0 1 0
0 0
,
0
0 1
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
and the rank of A = 3
Remark: To ﬁnd the rank of a matrix, the simplest method is to reduce to row echelon form.

Matrices ■ 1.23
EXERCISE 1.2
Find the rank of the following matrices reducing to echelon form.
1.
2 3 1 1
1 1 2 4
3 1 3 2
6 3 0 7
− −
− − −
−
−
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
2.
0 1 3 1
1 0 1 1
3 1 0 2
1 1 2 0
− −
−
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
3.
1 2 3 0
2 4 3 2
3 2 1 3
6 8 7 5
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
4.
4 3 0 2
3 4 1 3
7 7 1 5
−
−
− −
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
5.
1 2 1 3
4 1 2 1
3 1 1 2
1 2 0 1
−
−
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
6.
1 1 2 1
1 1 2 1
1 2 1 1
1 3 0 3
1 1 2 3
− − −
−
−
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
7.
1 1 1 1
1 3 2 1
2 0 3 2
3 3 3 3
−
−
−
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
8.
1 2 2 3
2 5 4 6
1 3 2 2
2 4 4 6
−
−
− − −
−
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
9.
3 1 5 1
1 2 1 5
1 5 7 2
− −
− −
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
10. Find the rank of the matrix A =
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
1 3 4 3
3 9 12 3
1 3 4 1
11. Find the rank of the matrix A =
− − −
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
2 1 1
12 8 6
10 5 6
.
12. Reduce the matrix
2 1 3 6
3 3 1 2
1 1 1 2
− −
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
to normal form and hence ﬁnd the rank.
13. Find the values of k if the rank of
4 4 3 1
1 1 1 0
2 2 2
9 9 3
−
−
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
k
k
is 3.
14. Find the values of a and b if the matrix
2 1 1 3
1 1 2 4
7 1
−
−
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
a b
is of rank 2.
15. Find the values of a and b if the matrix
1 2 3 1
2 1 1 2
6 2
−
−
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
a b
is of rank 2.
16. Reduce to normal form and ﬁnd the rank of
1 1 2 3
4 1 0 2
0 3 1 4
0 1 0 2
− −
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
.

17. Reduce to normal form and find the rank of
0 1 2 1
1 2 3 2
3 1 1 3
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
.
18. If A =
− −
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
1 1 2
1 2 3
0 1 1
, then find non-singular matrices P and Q such that PAQ is in normal form and find its
rank.
ANSWERS TO EXERCISE 1.2
1. 3 2. 2 3. 3 4. 2 5. 3
6. 3 7. 3 8. 3 9. 3 10. 2
11. 3 12. 3 13. k = 2 14. a = 4, b = 18 15. a = 4, b = 6
16. [I4
, 0], rank = 4 17. [I3
, 0], rank = 3
18. P Q
= −
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
=
− −
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
1 0 0
1 1 0
1 1 1
1 1 1
0 1 1
0 0 1
, and rank = 2.
1.4 SOLUTION OF SYSTEM OF LINEAR EQUATIONS
There are many problems in science and engineering whose solution often depends upon a system of
linear equations.
The equation a1
x1
+ a2
x2
+ … + an
xn
= b is called a non-homogeneous linear equation in n variables
x1
, x2
, …, xn
where b ≠ 0 and at least one ai
≠ 0.
If b = 0, then the equation a1
x1
+ a2
x2
+ … + an
xn
= 0 is called a homogenous linear equation in
x1
, x2
, …, xn
.
1.4.1 Non-homogeneous System of Equations
Consider the system of m linear equations in n variables x1
, x2
, …, xn
a11
x1
+ a12
x2
+ … + a1n
xn
= b1
a21
x1
+ a22
x2
+ … + a2n
xn
= b2
:
am1
x1
+ am2
x2
+ … + amn
xn
= bm
, where at least one bi
≠ 0
If A
a a a
a a a
a a a
B
b
b
b
n
n
m m mn m
=
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
=
⎡
11 12 1
21 22 2
1 2
1
2
…
…
…
: : : :
,
⎣
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
=
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
, ,
X
x
x
xn
1
2
:
then the system of equations can be written as a single matrix equation AX = B.
The matrix A is called the coefficient matrix.
A solution of the system is a set of values of x1
, x2
, …, xn
which satisfy the m equations.
The system of equations is said to be consistent if it has at least one solution.
If the system has no solution, then the system of equations is said to be inconsistent.
The condition for the consistency of the system is given by Rouche’s theorem.
We shall state the theorem without proof.

Matrices ■ 1.25
Theorem 1.2 Rouche’s Theorem The system of linear equations AX = B is consistent if and only if the
coefficient matrix A and the augmented matrix [A, B] have the same rank. That is., r(A) = r([A, B])
Working rule:
Let AX = B represent a system of m equations in n variables.
1. Write down the coefficient matrix A and the augmented matrix [A, B]. Find r(A), r([A, B])
2. If r(A) ≠ r([A, B]), then the system is inconsistent. That is it has no solution.
3. If r(A) = r([A, B]) = n, the number of variables, then the system is consistent with unique solution.
4. If r(A) = r([A, B]) n, the number of variables, then the system is consistent with infinite number
of solutions.
If the rank is r, then in this case the solution set will contain n − r parameters or arbitrary constants.
To get the solutions we assign arbitrary values to n − r variables and write down the solutions in
terms of them.
Forexample,thesystemx+y+z=1,2x−y+3z=−1,2x+5y+z=5isconsistentwithinfinitenumberof
solutions.
Here r = 2, n = 3.
∴ the solution set will contain n − r = 3 − 2 = 1 parameter.
We assign an arbitrary value to one variable, say y.
Put y = k and solve for x and z in terms of k.
The solution set is
x = 4 + 2k, y = k, z = −3 − 3k,
where k is any real number.
Note
If m = n, then A is a square matrix and the system of equations AX = B has unique solution if A is
non-singular. That is., A ≠ 0, then r(A) = number of variables n.
The unique solution is X = A−1
B.
1.4.2 Homogeneous System of Equations
Consider the homogeneous system
a11
x1
+ a12
x2
+ … + a1n
xn
= 0
a21
x1
+ a22
x2
+ … + a2n
xn
= 0
:
am1
x1
+ am2
x2
+ … + amn
xn
= 0
If A
a a a
a a a
a a a
X
x
x
x
n
n
m m mn n
=
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
=
11 12 1
21 22 2
1 2
1
2
…
…
…
: : : :
,
⎡
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
, then the matrix equation is AX = 0.
For this system x1
= 0, x2
= 0, …, xn
= 0 is always a solution. This is called the trivial solution.
If A ≠ 0, the r(A) = n and the only solution is the trivial solution. So, the condition for
non-trivial solution is A = 0 (or r(A) n).
In solving equations we use only row operations.

1.4.3 Type 1: Solution of Non-homogeneous System of Equations
WORKED EXAMPLES
(A) Non-homogeneous system with unique solution
EXAMPLE 1
Test for consistency and solve 2x 2 y 1 z 5 7, 3x 1 y 2 5z 5 13, x 1 y 1 z 5 5.
Solution.
The given equations are
x + y + z = 5
2x − y + z = 7
3x + y − 5z = 13
We have rearranged the equation for convenience in reducing to row echelon form.
The coefﬁcient matrix is A = −
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
1 1 1
2 1 1
3 1 5
and the augmented matrix is
[ , ]
:
:
:
:
:
:
A B = −
−
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
− − −
− − −
⎡
⎣
1 1 1 5
2 1 1 7
3 1 5 13
1 1 1 5
0 3 1 3
0 2 8 2
∼ ⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
→ −
→ −
R R R
R R R
2 2 1
3 3 1
2
3
∼
∼
1 1 1 5
0 1
1
3
1
0 1 4 1
1
3
1
2
1 1 1 5
0 1
1
3
2 2
3 3
:
:
:
:
:
− − −
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
→ −
→
R R
R R
1
1
0 0
11
3
0 3 3 2
−
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
→ +
: R R R
From the last matrix, we ﬁnd
A ∼
1 1 1
0 1
1
3
0 0
11
3
−
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥

Matrices ■ 1.27
The number of non-zero rows in the equivalent matrices of A and [A, B] are 3.
∴ r(A) = 3, r([A, B]) = 3
⇒ r(A) = r([A, B]) = 3, the number of variables.
So, the equations are consistent with unique solution.
From the reduced matrix [A, B], we ﬁnd the given equations are equivalent to
x + y + z = 5, y +
1
3
z = 1 and −
11
3
z = 0 ⇒ z = 0
∴ y = 1 and x + 1 + 0 = 5 ⇒ x = 4.
So, the unique solution is x = 4, y = 1, z = 0.
EXAMPLE 2
Test for the consistency and solve x 1 2y 1 z 5 3, 2x 1 3y 1 2z 5 5, 3x 2 5y 1 5z 5 2,
3x 1 9y 2 z 5 4.
Solution.
x + 2y + z = 3
2x + 3y + 2z = 5
3x − 5y + 5z = 2
3x + 9y − z = 4.
The coefﬁcient matrix is A =
−
−
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
1 2 1
2 3 2
3 5 5
3 9 1
The augmented matrix is
[ , ]
:
:
:
:
:
:
A B = ∼
1 2 1 3
2 3 2 5
3 5 5 2
3 9 1 4
1 2 1 3
0 1 0 1
0 11 2
−
−
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
− −
− :
:
:
:
−
− −
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
→ −
→ −
→ −
−
7
0 3 4 5
2
3
3
1 2 1 3
0
2 2 1
3 3 1
4 4 1
R R R
R R R
R R R
∼
1
1 0 1
0 0 2 4
0 0 4 8
11
3
1 2 1 3
0 1
3 3 2
4 4 2
:
:
:
:
−
− −
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
→ −
→ +
−
R R R
R R R
∼
0
0 1
0 0 1 2
0 0 1 2
1
2
1
4
3 3
4 4
:
:
:
−
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
→
→ −
R R
R R

⇒ [ , ]
:
:
:
:
A B
R R R
∼
1 2 1 3
0 1 0 1
0 0 1 2
0 0 0 0 4 4 3
− −
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
→ −
From this last matrix we find
A ∼
1 2 1
0 1 0
0 0 1
0 0 0
−
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
The number of non-zero rows in the equivalent matrices of A and [A B] are 3.
∴ r(A) = 3, r([A, B]) = 3
So, the equations are consistent with unique solution
From the reduced matrix [A, B], we find the given equations are equivalent to
x + 2y + z = 3, −y = −1 and z = 2
∴ y = 1, z = 2 and so x + 2 ⋅ 1 + 2 = 3 ⇒ x = −1
So, the unique solution is x = −1, y = 1, z = 2.
EXAMPLE 3
Solve x2
yz 5 e, xy2
z3
5 e, x3
y2
z 5 e using matrices.
Solution.
x2
yz = e (1)
xy2
z3
= e (2)
x3
y2
z = e (3)
Taking logarithm to the base e on both sides of (1), (2) and (3), we get
and
log log log log log
log log log
lo
e e e e e
e e e
x yz e x y z
x y z
2 2
1
2 1
= ⇒ + + =
⇒ + + =
g
g log log log log
log log log
e e e e e
e e
xy z e x y z
x y z e x
2 3
3 2
2 3 1
3
= ⇒ + + =
= ⇒ + 2
2 1
log log
e e
y z
+ =
For simplicity, put x1
= loge
x, y1
= loge
y, z1
= loge
z
∴ the equations are 2x1
+ y1
+ z1
= 1
x1
+ 2y1
+ 3z1
= 1
3x1
+ 2y1
+ z1
= 1
The coefficient matrix is A =
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
2 1 1
1 2 3
3 2 1

Matrices ■ 1.29
The augmented matrix is
[ , ]
:
:
:
A B =
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
2 1 1 1
1 2 3 1
3 2 1 1
∼
∼
1 2 3 1
2 1 1 1
3 2 1 1
1 2 3 1
0 3 5 1
0 4 8 2
1 2
:
:
:
:
:
:
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
↔
− − −
− − −
⎡
⎣
⎢
⎢
⎢
R R
⎤
⎤
⎦
⎥
⎥
⎥
→ −
→ −
R R R
R R R
2 2 1
3 3 1
2
3
⇒ − − −
− −
⎡
⎣
⎢
⎢
⎢
⎢
A B
1 2 3 1
0 3 5 1
0 0
4
3
2
3
[ , ]
:
:
:
∼
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥ → −
R R R
3 3 2
4
3
From the last matrix, we ﬁnd A ∼
1 2 3
0 3 5
0 0
4
3
− −
−
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
The number of non-zero rows in the equivalent matrices of A and [A, B] are 3.
∴ r(A) = 3, r([A, B]) = 3
So, the equations are consistent with unique solution.
From the reduced matrix [A, B], we ﬁnd that the given equations are equivalent to
x1
+ 2y1
+ 3z1
= 1 (4)
−3y1
− 5z1
= −1 (5)
and − = − ⇒ =
4
3
2
3
1
2
1 1
z z
Substituting in (5), we get
− − ⋅ = − ⇒ = − = − ⇒ = −
3 5
1
2
1 3 1
5
2
3
2
1
2
1 1 1
y y y
Substituting in (4) we get
⇒
∴
x x x
x x x e
e
1 1 1
1
1
2
1
2
3
1
2
1
1
2
1 1
1
2
1
2
1
2
1
2
+ −
⎛
⎝
⎜
⎞
⎠
⎟ + ⋅ = ⇒ + = ⇒ = − =
= ⇒ = ⇒ =
log 2
2
1
1
2
1
1
2
1
2
1
2
1
1
2
1
2
=
= − ⇒ = − ⇒ = =
= ⇒ = ⇒ = =
−
e
y y y e
e
z z z e e
e
e
log
log

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