P_Sivaramakrishna_Das,_C_Vijayakumari_Engineering_Mathematics_Pearson (2)

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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
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Copyright © 2017 Pearson India Education Services Pvt. Ltd
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eISBN 978-93-325-8776-2
 
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Preface xxxi
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
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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
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Preface xxxi
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
 Worked Examples 1.16
 Exercise 1.2 1.23
 Answers to Exercise 1.2 1.24
 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 Examples 1.26
 1.4.4 Type 2: Solution of Non-homogeneous Linear Equations Involving 
Arbitrary Constants 1.34
 Worked Examples 1.34
 1.4.5 Type 3: Solution of the System of Homogeneous Equations 1.38
 Worked Examples 1.38
 1.4.6 Type 4: Solution of Homogeneous System of Equation Containing 
Arbitrary Constants 1.41
 Worked Examples 1.41
 Exercise 1.3 1.44
 Answers to Exercise 1.3 1.45
 1.5 Matrix Inverse by Gauss–Jordan method 1.46
 Worked Examples 1.47
 Exercise 1.4 1.53
 Answers to Exercise 1.4 1.53
Contents
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x n Contents
 1.6 Eigen Values and Eigen Vectors 1.54
 1.6.0 Introduction 1.54
 1.6.1 Vector 1.54
 Worked Examples 1.55
 1.6.2 Eigen Values and Eigen Vectors 1.56
 1.6.3 Properties of Eigen Vectors 1.57
 Worked Examples 1.58
 1.6.4 Properties of Eigen Values 1.67
 Worked Examples 1.70
 Exercise 1.5 1.72
 Answers to Exercise 1.5 1.73
 1.6.5 Cayley-Hamilton Theorem 1.73
 Worked Examples 1.75
 Exercise 1.6 1.82
 Answers to Exercise 1.6 1.83
 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
 Worked Examples 1.90
 1.8 Real Quadratic Form. Reduction to Canonical Form 1.96
 Worked Examples 1.99
 Exercise 1.7 1.111
 Answers to Exercise 1.7 1.112
 Short Answer Questions 1.113
 Objective Type Questions 1.114
 Answers 1.116
2. Sequences and Series 2.1
 2.0 Introduction 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
 Answers to Exercise 2.1 2.9
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Contents n xi
 2.2 Series 2.9
 2.2.1 Convergent Series 2.9
 2.2.2 Divergent Series 2.10
 2.2.3 Oscillatory Series 2.10
 2.2.4 General Properties of Series 2.10
 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 Series 2.11
 2.3.3 Comparison Tests 2.11
 Worked Examples 2.13
 Exercise 2.2 2.17
 Answers to Exercise 2.2 2.18
 2.3.4 De’ Alembert’s Ratio Test 2.18
 Worked Examples 2.21
 Exercise 2.3 2.25
 Answers to Exercise 2.3 2.26
 2.3.5 Cauchy’s Root Test 2.27
 Worked Examples 2.28
 2.3.6 Cauchy’s Integral Test 2.30
 Worked Examples 2.32
 Exercise 2.4 2.36
 Answers to Exercise 2.4 2.36
 2.3.7 Raabe’s Test 2.36
 Worked Examples 2.37
 Exercise 2.5 2.41
 Answers to Exercise 2.5 2.42
 2.3.8 Logarithmic Test 2.42
 Worked Examples 2.44
 2.4 Alternating Series 2.46
 2.4.1 Leibnitz’s Test 2.46
 Worked Examples 2.47
 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 Convergence 2.50
 Worked Examples 2.51
 Exercise 2.6 2.55
 Answers to Exercise 2.6 2.55
 2.6 Convergence of Binomial Series 2.56
 2.7 Convergence of the Exponential Series 2.572.8 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 Series 2.60
 Worked Examples 2.60
 Exercise 2.7 2.66
 Answers to Exercise 2.7 2.67
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 Short Answer Questions 2.67
 Objective Type Questions 2.69
 Answers 2.70
3. Differential Calculus 3.1
 3.0 Introduction 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
 Worked Examples 3.11
 Exercise 3.2 3.16
 Answers to Exercise 3.2 3.17
 Worked Examples 3.18
 Exercise 3.3 3.24
 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 Examples 3.26
 Exercise 3.4 3.33
 Answers to Exercise 3.4 3.34
 3.2.3 Length of the Tangent, the Sub-Tangent, the Normal and the Sub-normal 3.34
 Worked Examples 3.36
 Exercise 3.5 3.38
 Answers to Exercise 3.5 3.38
 3.2.4 Angle between the Two Curves 3.38
 Worked Examples 3.39
 Exercise 3.6 3.42
 Answers to Exercise 3.6 3.43
 3.3 Mean-value Theorems of Derivatives 3.43
 3.3.1 Rolle’s Theorem 3.43
 Worked Examples 3.44
 3.3.2 Lagrange’s Mean Value Theorem 3.47
 Worked Examples 3.49
 3.3.3 Cauchy’s Mean Value Theorem 3.53
 Worked Examples 3.54
 Exercise 3.7 3.56
 Answers to Exercise 3.7 3.58
 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 Functions 3.59
 Worked Examples 3.60
 Exercise 3.8 3.65
 Answers to Exercise 3.8 3.66
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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
 Worked Examples 3.69
 Exercise 3.9 3.74
 Answers to Exercise 3.9 3.74
 3.5.5 Expansion by Using Maclaurin’s Series of Some Standard Functions 3.75
 Worked Examples 3.75
 3.5.6 Expansion of Certain Functions Using Differential Equations 3.78
 Worked Examples 3.78
 Exercise 3.10 3.81
 Answers to Exercise 3.10 3.82
 3.6 Indeterminate Forms 3.82
 3.6.1 General L’Hopital’s Rule for 
0
0
 Form 3.84
 Worked Examples 3.85
 Exercise 3.11 3.94
 Answers to 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
 Worked Examples 3.97
 Exercise 3.12 3.103
 Answers to Exercise 3.12 3.104
 3.8 Asymptotes 3.104
 Worked Examples 3.105
 3.8.1 A General Method 3.108
 3.8.2 Asymptotes Parallel to the Coordinates Axes 3.110
 Worked Examples 3.110
 3.8.3 Another Method for Finding the Asymptotes 3.113
 Worked Examples 3.114
 3.8.4 Asymptotes by Inspection 3.115
 Worked Examples 3.116
 3.8.5 Intersection of a Curve and Its Asymptotes 3.116
 Worked Examples 3.116
 Exercise 3.13 3.121
 Answers to Exercise 3.13 3.122
 3.9 Concavity 3.122
 Worked Examples 3.124
 Exercise 3.14 3.127
 Answers to Exercise 3.14 3.128
 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
 Worked Examples 3.129
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 3.10.2 Procedure for Tracing of Curve Given by Parametric Equations 
x = f(t), y = g(t) 3.137
 Worked Examples 3.137
 3.10.3 Procedure for Tracing of Curve Given by Equation in Polar 
Coordinates f(r, u) = 0 3.141
 Worked Examples 3.142
 Exercise 3.15 3.146
 Answers to Exercise 3.15 3.146
 Short Answer Questions 3.148
 Objective Type Questions 3.149
 Answers 3.152
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
 Worked Examples 4.13
 Exercise 4.1 4.15
 Answers to Exercise 4.1 4.16
 4.1.6 Radius of Curvature in Polar Coordinates 4.17
 Worked Examples 4.19
 4.1.7 Radius of Curvature at the Origin 4.22
 Worked Examples 4.23
 4.1.8 Pedal Equation or p - r Equation of a Curve 4.25
 Worked Examples 4.26
 4.1.9 Radius of Curvature Using the p - r Equation of a Curve 4.28
 Worked Examples 4.29
 Exercise 4.2 4.30
 Answers to Exercise 4.2 4.31
 4.2 Evolute 4.31
 4.2.1 Properties of Evolute 4.31
 4.2.2 Procedure to Find the Evolute 4.34
 Worked Examples 4.34
 Exercise 4.3 4.41
 Answers to 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
 Worked Examples 4.43
 4.3.2 Envelope of Two Parameter Family of Curves 4.45
 Worked Examples 4.45
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Contents n xv
 4.3.3 Evolute as the Envelope of Normals 4.48
 Worked Examples 4.49
 Exercise 4.4 4.52
 Answers to Exercise 4.4 4.53
Short Answer Questions 4.54
Objective Type Questions 4.54
Answers 4.56
5. Differential Calculus of Several Variables 5.1
 5.0 Introduction 5.1
 5.1 Limit and Continuity 5.1
 Worked Examples 5.4
 Exercise 5.1 5.6
 Answers to 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
 Worked Examples 5.17
 Exercise 5.2 5.24
 Answers to Exercise 5.2 5.26
 5.3 Jacobians 5.26
 5.3.1 Properties of Jacobians 5.27
 Worked Examples 5.29
 5.3.2 Jacobian of Implicit Functions 5.35
 Worked Examples 5.35
 Exercise 5.3 5.37
 Answers to Exercise 5.3 5.38
 5.4 Taylor’s Series Expansion for Function of Two Variables 5.38
 Worked Examples 5.39
 Exercise 5.4 5.44
 Answers to Exercise 5.4 5.44
 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 Examples 5.47
 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 Examples 5.56
 Exercise 5.5 5.65
 Answers to Exercise 5.5 5.66
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 5.6 Errors and Approximations 5.67
 Worked Examples 5.68
 Exercise 5.6 5.72
 Answers to Exercise 5.6 5.73
Short Answer Questions 5.73
Objective Type Questions 5.74
Answers 5.76
6. Integral Calculus 6.1
 6.0 Introduction 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
 Answers to Exercise 6.1 6.9
 6.2 Definite Integral (Newton–Leibnitz formula) 6.10
 6.2.1 Properties of Definite Integral 6.10
 Worked Examples 6.15
 Exercise 6.2 6.27
 Answers to Exercise 6.2 6.27
 6.3 Definite Integral f x dx
a
b
( )∫ as Limit of a Sum 6.28
 6.3.1 Working Rule 6.28
 Worked Examples 6.29
 Exercise 6.3 6.32
 Answers to Exercise 6.3 6.33
 6.4 Reduction Formulae 6.33
 6.4.1 The Reduction Formula for (a) sin
n x dx∫ and (b) cosn x dx∫ 6.33
 6.4.2 The Reduction Formula for (a) tann x dx∫ and (b) cotn x dx∫ 6.36
 6.4.3 The Reduction Formula for (a) secn x dx∫ and (b) cosecn x dx∫ 6.37
 Worked Examples 6.38
 6.4.4 The Reduction Formula for sin cosm nx x dx∫ , Where m, n are 
Non-negative Integers 6.45
 Worked Examples6.47
 6.4.5 The Reduction Formula for (a) ∫xm(log x)ndx, (b) ∫ xn sin mx dx, 
(c) ∫ xn cos mx dx 6.49
 6.4.6 The Reduction Formula for (a) e x dx
ax msin∫ and (b) e x dxax ncos∫ 6.51
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Contents n xvii
 6.4.7 The Reduction Formula for (a) cos sinm x nx dx∫ and (b) cos cosm x nx dx∫ 6.52
 Exercise 6.4 6.55
 Answers to 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
Worked Examples 6.57
Exercise 6.5 6.66
Answers to Exercise 6.5 6.67
6.5.1 (b) Area in Polar Coordinates 6.67
Worked Examples 6.68
Exercise 6.6 6.72
Answers to 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
 Worked Examples 6.73
 Exercise 6.7 6.78
 Answers to Exercise 6.7 6.79
6.5.2 (b) Length of the Arc in Polar Coordinates 6.79
 Worked Examples 6.80
 Exercise 6.8 6.81
 Answers to 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
 Worked Examples 6.83
 Exercise 6.9 6.89
 Answers to Exercise 6.9 6.90
6.5.3 (b) Volume in Polar Coordinates 6.91
 Worked Examples 6.91
 Exercise 6.10 6.93
 Answers to 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
 Worked Examples 6.94
 Exercise 6.11 6.99
 Answers to Exercise 6.11 6.99
6.5.4 (b) Surface Area in Polar Coordinates 6.100
 Worked Examples 6.100
 Exercise 6.12 6.102
 Answers to Exercise 6.12 6.103
Short Answer Questions 6.103
Objective Type Questions 6.103
Answers 6.106
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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 7.1 7.13
 Answers to Exercise 7.1 7.13
 7.1.2 Tests of Convergence of Improper Integrals 7.14
 Worked Examples 7.15
 Exercise 7.2 7.27
 Answers to Exercise 7.2 7.27
 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 Examples 7.28
 Exercise 7.3 7.47
 Answers to 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 Function 7.48
 7.3.3 Different Forms of Beta Function 7.48
 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 Examples 7.55
 Exercise 7.4 7.69
 Answers to Exercise 7.4 7.69
 7.5 The Error Function 7.70
 7.5.1 Properties of Error Functions 7.70
 7.5.2 Series Expansion for Error Function 7.71
 7.5.3 Complementary Error Function 7.71
 Worked Examples 7.72
 Exercise 7.5 7.76
 Answers to Exercise 7.5 7.76
Short Answer Questions 7.76
Objective Type Questions 7.77
Answers 7.78
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
 Answers to Exercise 8.1 8.7
 8.1.3 Change of Order of Integration 8.7
 Worked Examples 8.8
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Contents n xix
 Exercise 8.2 8.15
 Answers to Exercise 8.2 8.15
 8.1.4 Double Integral in Polar Coordinates 8.16
 Worked Examples 8.16
 8.1.5 Change of Variables in Double Integral 8.19
 Worked Examples 8.19
 Exercise 8.3 8.26
 Answers to Exercise 8.3 8.27
 8.1.6 Area as Double Integral 8.27
 Worked Examples 8.28
 Exercise 8.4 8.31
 Answers to Exercise 8.4 8.31
 Worked Examples 8.32
 Exercise 8.5 8.37
 Answers to Exercise 8.5 8.37
 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 Examples 8.41
 Exercise 8.6 8.49
 Answers to Exercise 8.6 8.49
 8.3 Triple Integral in Cartesian Coordinates 8.49
 Worked Examples 8.50
 Exercise 8.7 8.55
 Answers to Exercise 8.7 8.56
 8.3.1 Volume as Triple Integral 8.56
 Worked Examples 8.56
 Exercise 8.8 8.63
 Answers to Exercise 8.8 8.64
Short Answer Questions 8.64
Objective Type Questions 8.64
Answers 8.66
9. Vector Calculus 9.1
 9.0 Introduction 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
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 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
 Answers to Exercise 9.1 9.21
 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
 Worked Examples 9.24
 Exercise 9.2 9.30
 Answers to Exercise 9.2 9.31
 9.7 Vector Identities 9.31
 Worked Examples 9.37
 9.8 Integration of Vector Functions 9.39
 9.8.1 Line Integral 9.40
 Worked Examples 9.40
 Exercise 9.3 9.46
 Answers to Exercise 9.3 9.47
 9.9 Green’s Theorem in a Plane 9.47
 9.9.1 Vector Form of Green’s Theorem 9.50
 Worked Examples 9.50
 9.10 Surface Integrals 9.56
 9.10.1 Evaluation of Surface Integral 9.57
 9.11 Volume Integral 9.58
 Worked Examples 9.58
 9.12 Gauss Divergence Theorem 9.62
 9.12.1 Results Derived from Gauss Divergence Theorem 9.64
 Worked Examples 9.68
 9.13 Stoke’s Theorem 9.81
 Worked Examples 9.83
 Exercise 9.4 9.97
 Answers to Exercise 9.4 9.100
Short Answer Questions 9.100
Objective Type Questions 9.101
Answers 9.102
10. Ordinary First Order Differential Equations 10.1
 10.0 Introduction 10.1
 10.1 Formation of Differential Equations 10.2
 Worked Examples 10.2
 Exercise 10.1 10.5
 Answers to Exercise 10.1 10.6
 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
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Contents n xxi
 Worked Example 10.6
 Exercise 10.2 10.9
 Answers to Exercise 10.2 10.9
 10.2.2 Type II
 Homogeneous Equation 10.10
 Worked Examples 10.10
 Exercise 10.3 10.13
 Answers to Exercise 10.3 10.14
 10. 2.3 Type III
 Non-Homogenous Differential Equations of the First Degree 10.14
 Worked Examples 10.16
 Exercise 10.4 10.21
 Answers to Exercise 10.4 10.21
 10.2.4 Type IV
 Linear Differential Equation 10.22
 Worked Examples 10.23
 Exercise 10.5 10.27
 Answers to Exercise 10.5 10.27
 10.2.5 Type V Bernoulli’s Equation 10.28
 Worked Examples 10.28
 Exercise 10.6 10.31
 Answers to Exercise 10.6 10.31
 10.2.6 Type VI Riccati Equation 10.31
 Worked Examples 10.33
 Exercise 10.7 10.36
 Answers to Exercise 10.7 10.36
 10.2.7 Type VII First Order Exact Differential Equations 10.37
 Worked Examples 10.39
 Exercise 10.8 10.41
 Answers to Exercise 10.8 10.42
 10.3 Integrating Factors 10.42
 Worked Examples 10.43
 10.3.1 Rules for Finding the Integrating Factor for Non-Exact Differential 
Equation Mdx + Ndy = 0 10.45
 Worked Examples 10.46
 Exercise 10.9 10.56
 Answers to Exercise 10.9 10.56
 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
 Worked Examples 10.57
 Exercise 10.10 10.59
 Answers to Exercise 10.10 10.60
 10.4.2 Type 2
 Equations Solvable for y 10.60
 Worked Examples 10.61
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xxii n Contents
 10.4.3 Type 3
 Equations Solvable for x 10.64
 Worked Examples 10.65
 Exercise 10.11 10.67
 Answers to Exercise 10.11 10.67
 10.4.4 Type 4 Clairaut’s Equation 10.67
 Worked Examples 10.68
 Exercise 10.12 10.71
 Answers to Exercise 10.12 10.71
Short Answer Questions 10.71
Objective Type Questions 10.72
Answers 10.74
11. Ordinary Second and Higher Order Differential Equations 11.1
 11.0 Introduction 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
 Worked Examples 11.3
 Exercise 11.1 11.19
 Answers to Exercise 11.1 11.19
 11.2 Linear Differential Equations with Variable Coefficients 11.21
 11.2.1 Cauchy’s Homogeneous Linear Differential Equations 11.21
 Worked Examples 11.22
 11.2.2 Legendre’s Linear Differential Equation 11.29
 Worked Examples 11.30
 Exercise 11.2 11.32
 Answers to Exercise 11.2 11.33
 11.3 Simultaneous Linear Differential Equations with 
Constant Coefficients 11.34
Worked Examples 11.34
Exercise 11.3 11.43
Answers to Exercise 11.3 11.44
 11.4 Method of Variation of Parameters 11.44
 11.4.1 Working Rule 11.45
Worked Examples 11.45
Exercise 11.4 11.51
Answers to Exercise 11.4 11.52
 11.5 Method of Undetermined Coefficients 11.52
Worked Examples 11.54
Exercise 11.5 11.60
Answers to Exercise 11.5 11.60
Short Answers Questions 11.60
Objective Type Questions 11.61
Answers 11.63
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Contents n xxiii
12. Applications of Ordinary Differential Equations 12.1
 12.0 Introduction 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
 Worked Examples 12.2
 Exercise 12.1 12.7
 Answers To Exercise 12.1 12.8
 12.1.3 Chemical Reaction and Solutions 12.8
 Worked Examples 12.9
 Exercise 12.2 12.12
 Answers to Exercise 12.2 12.13
 12.1.4 Simple Electric Circuit 12.13
 Worked Examples 12.14
 Exercise 12.3 12.19
 Answers to Exercise 12.3 12.19
 12.1.5 Geometrical Applications 12.20
 12.1.5 (a) Orthogonal Trajectories in Casterian Coordinates 12.20
 Worked Examples 12.21
 12.1.5 (b) Orthogonal Trajectories in Polar Coordinates 12.23
 Worked Examples 12.24
 Exercise 12.4 12.26
 Answers to Exercise 12.4 12.27
 12.2 Applications of Second Order Differential Equations 12.27
 12.2.1 Bending of Beams 12.27
 Worked Examples 12.29
 12.2.2 Electric Circuits 12.34
 Worked Examples 12.34
 Exercise 12.5 12.38
 Answers to Exercise 12.5 12.39
 12.2.3 Simple Harmonic Motion (S.H.M) 12.40
 Worked Examples 12.41
 Exercise 12.6 12.43
 Answers to Exercise 12.6 12.44
Objective Type Questions 12.44
Answers 12.45
13. Series Solution of Ordinary Differential Equations and Special Functions 13.1
 13.0 Introduction 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
 Worked Examples 13.3
 Exercise 13.1 13.9
 Answers to Exercise 13.1 13.9
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xxiv n Contents
 13.2 Frobenius Method 13.9
 Worked Examples 13.11
 Exercise 13.2 13.33
 Answers to Exercise 13.2 13.33
 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
 Worked Examples 13.39
 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
 Worked Examples 13.46
 13.4.6 Integral Formula for Bessel’s Function Jn (x) 13.49
 Worked Examples 13.53
 13.4.7 Orthogonality of Bessel’s Functions 13.56
 13.4.8 Fourier–Bessel Expansion of a Function f(x) 13.59
 Worked Examples 13.60
 13.4.9 Equations Reducible to Bessel’s Equation 13.62
 Worked Examples 13.62
 Exercise 13.3 13.65
 Answers to Exercise 13.3 13.66
 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
 Worked Examples 13.73
 13.5.4 Generating Function for Legendre Polynomials 13.74
 Worked Examples 13.75
 13.5.5 Orthogonality of Legendre Polynomials in [-1, 1] 13.77
 Worked Examples 13.80
 13.5.6 Fourier–Legendre Expansion of f(x) in a Series of Legendre 
Polynomials 13.83
 Worked Examples 13.83
 Exercise 13.4 13.85
 Answers to Exercise 13.4 13.85
14. Partial Differential Equations 14.1
 14.0 Introduction 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
 Worked Examples 14.2
 Exercise 14.1 14.15
 Answers to Exercise 14.1 14.15
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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 Equation 14.17
 Worked Examples 14.17
 Exercise 14.2 14.20
 Answers to Exercise 14.2 14.20
 14.4.2 First Order Non-linear Partial Differential Equation of Standard Types 14.20
 Worked Examples 14.21
 Exercise 14.3 14.25
 Answers to Exercise 14.3 14.25
 Worked Examples 14.26
 14.4.3 Equations Reducible to Standard Forms 14.33
 Worked Examples 14.35
 Exercise 14.4 14.38
 Answers to Exercise 14.4 14.38
 14.5 Lagrange’s Linear Equation 14.39
 Worked Examples 14.41
 Exercise 14.5 14.48
 Answers to Exercise 14.5 14.48
 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
 Worked Examples 14.53
 Exercise 14.6 14.66
 Answers to Exercise 14.6 14.67
 14.7 Non-homogeneous Linear Partial Differential Equations of the 
Second and Higher Order with Constant Coefficients 14.68
 Worked Examples 14.69
 Exercise 14.7 14.73
 Answers to Exercise 14.7 14.73
Short Answer Questions 14.74
Objective Type Questions 14.74
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
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 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 Form 15.10
 Worked Examples 15.11
 Exercise 15.1 15.20
 Answers to Exercise 15.1 15.21
 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
 Worked Examples 15.25
 Exercise 15.2 15.32
 Answers to Exercise 15.2 15.33
 15.5 Conformal Mapping 15.33
 15.5.1 Angle of Rotation 15.34
 15.5.2 Mapping by Elementary Functions 15.36
 Worked Examples 15.37
 Exercise 15.3 15.72
 Answers to Exercise 15.3 15.74
 15.5.3 Bilinear Transformation 15.79
 Worked Examples 15.82
 Exercise 15.4 15.89
 Answers to Exercise 15.4 15.90
Short Answer Questions 15.90
Objective Type Questions 15.91
Answers 15.92
16. Complex Integration 16.1
 16.0 Introduction 16.1
 16.1 Contour Integral 16.1
 16.1.1 Properties of Contour Integrals 16.1
 Worked Examples 16.2
 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’sIntegral Formula for Derivatives 16.7
 Worked Examples 16.7
 Exercise 16.1 16.12
 Answers to Exercise 16.1 16.13
 16.4 Taylor’s Series and Laurent’s Series 16.14
 16.4.1 Taylor’s Series 16.14
 16.4.2 Laurent’s Series 16.15
 Worked Examples 16.16
 Exercise 16.2 16.22
 Answers to Exercise 16.2 16.23
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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
 Worked Examples 16.28
 Exercise 16.3 16.34
 Answers to Exercise 16.3 16.36
 16.8 Application of Residue Theorem to Evaluate Real Integrals 16.36
 16.8.1 Type 1 16.36
 Worked Examples 16.37
 16.8.2 Type 2. Improper Integrals of Rational Functions 16.44
 Worked Examples 16.46
 16.8.3 Type 3 16.50
 Worked Examples 16.50
 Exercise 16.4 16.55
 Answers to Exercise 16.4 16.56
Short Answer Questions 16.56
Objective Type Questions 16.58
Answers 16.60
17 Fourier Series 17.1
 17.0 Introduction 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
 Worked Examples 17.5
 17.2 Even and Odd Functions 17.15
 17.2.1 Sine and Cosine Series 17.15
 Worked Examples 17.16
 Exercise 17.1 17.23
 Answers to Exercise 17.1 17.25
 17.3 Half-Range Series 17.26
 17.3.1 Half-range Sine Series 17.27
 17.3.2 Half-range Cosine Series 17.27
 Worked Examples 17.28
 Exercise 17.2 17.36
 Answers to Exercise 17.2 17.37
 17.4 Change of Interval 17.38
 Worked Examples 17.39
 17.5 Parseval’s Identity 17.47
 Worked Examples 17.47
 Exercise 17.3 17.50
 Answers to Exercise 17.3 17.52
 17.6 Complex Form of Fourier Series 17.53
 Worked Examples 17.55
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xxviii n Contents
 Exercise 17.4 17.59
 Answers to Exercise 17.4 17.59
 17.7 Harmonic Analysis 17.60
 17.7.1 Trapezoidal Rule 17.60
 Worked Examples 17.62
 Exercise 17.5 17.68
 Answers to Exercise 17.5 17.69
Short Answer Questions 17.69
Objective Type Questions 17.70
Answers 17.72
18. Fourier Transforms 18.1
 18.0 Introduction 18.1
 18.1 Fourier Integral Theorem 18.1
 18.1.1 Fourier Cosine and Sine Integrals 18.2
 Worked Examples 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
 Worked Examples 18.12
 Exercise 18.1 18.21
 Answers to Exercise 18.1 18.22
 18.3 Fourier Sine and Cosine Transforms 18.23
 18.3.1 Properties of Fourier Sine and Cosine Transforms 18.24
 Worked Examples 18.29
 Exercise 18.2 18.39
 Answers to Exercise 18.2 18.39
 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 Theorem 18.41
 18.4.3 Theorem 18.2 : Parseval’s Identity for Fourier Transforms or 
Energy Theorem 18.41
 Worked Examples 18.43
 Exercise 18.3 18.51
 Answers to Exercise 18.3 18.52
Short Answer Questions 18.52
Objective Type Questions 18.53
Answers 18.54
19. Laplace Transforms 19.1
 19.0 Introduction 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
 Worked Examples 19.5
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Contents n xxix
 Exercise 19.1 19.9
 Answers to Exercise 19.1 19.10
 19.4 Differentiation and Integration of Transforms 19.11
 Worked Examples 19.12
 Exercise 19.2 19.20
 Answers to Exercise 19.2 19.20
 19.5 Laplace Transform of Derivatives and Integrals 19.21
 Worked Examples 19.23
 19.5.1 Evaluation of Improper Integrals using Laplace Transform 19.25
 Worked Examples 19.25
 19.6 Laplace Transform of Periodic Functions and Other Special Type of Functions 19.27
 Worked Examples 19.29
 19.6.1 Laplace Transform of Unit Step Function 19.36
 19.6.2 Unit Impulse Function 19.37
 19.6.3 Dirac-delta Function 19.37
 19.6.4 Laplace Transform of Delta Function 19.37
 Worked Examples 19.38
 Exercise 19.3 19.39
 Answers to Exercise 19.3 19.41
 19.7 Inverse Laplace Transforms 19.41
 19.7.1 Type 1 – Direct and Shifting Methods 19.43
 Worked Examples 19.43
 19.7.2 Type 2 – Partial Fraction Method 19.44
 Worked Examples 19.44
 19.7.3 Type 3 – 1. Multiplication by s and 2. Division by s 19.48
 Worked Examples 19.48
 19.7.4 Type 4 – Inverse Laplace Transform of Logarithmic and 
Trigonometric Functions 19.50
 Worked Examples 19.50
 Exercise 19.4 19.53
 Answers to Exercise 19.4 19.54
 19.7.5 Type 5 – Method of Convolution 19.55
 Worked Examples 19.57
 Exercise 19.5 19.60
 Answers to Exercise 19.5 19.61
 19.7.6 Type 6: Inverse Laplace Transform as Contour Integral 19.61
 Worked Examples 19.62
 Exercise 19.6 19.64
 Answers to Exercise 19.6 19.65
 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
 Worked Examples 19.65
 19.8.2 Ordinary Second and Higher Order Linear Differential Equations 
with Constant Coefficients 19.68
 Worked Examples 19.68
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 19.8.3 Ordinary Second Order Differential Equations with Variable Coefficients 19.72
 Worked Examples 19.72
 Exercise 19.7 19.75
 Answers to Exercise 19.7 19.76
 19.8.4 Simultaneous Differential Equations 19.77
 Worked Examples 19.77
 19.8.5 Integral–Differential Equation 19.83
 Worked Examples 19.83
 Exercise 19.8 19.85
 Answers to Exercise 19.8 19.86
Short Answer Questions 19.86
Objective Type Questions 19.86
Answers 19.88
20. Applications of Partial Differential Equations 20.1
 20.0 Introduction 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
 Worked Examples 20.5
 Exercise 20.1 20.34
 Answers to Exercise 20.1 20.35
 20.1.3 Classification of Partial Differential Equation of Second Order 20.36
 Worked Examples 20.37
 Exercise 20.2 20.38
 Answers to Exercise 20.2 20.38
 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
 Worked Examples 20.42
 Exercise 20.3 20.62
 Answers to Exercise 20.3 20.63
 Worked Examples 20.64
 Exercise 20.4 20.68
 Answers to Exercise 20.4 20.69
 20.3 Two Dimensional Heat Equation in Steady State 20.69
 20.3.1 Solution of Two Dimensional Heat Equation 20.70
 Worked Examples 20.71
 Exercise 20.5 20.83
 Answers to Exercise 20.5 20.84
Short Answer Questions 20.85
Objective Type Questions 20.86
Answers 20.88
Index I.1
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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
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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
A01_ENGINEERING_MATHEMATICS-I _FM - (Reprint).indd 32 3/2/2017 6:17:56 PM
mailto:profpsdas@yahoo.com
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
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Engineering Mathematics
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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
AA A A
a a a a am m m mj mn1 2 3 … …
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
where a
ij
 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 = [a
ij
]
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 squarematrix of order n, then A = [a
ij
]
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.
1Matrices
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 = [a
11
 a
12
 a
13
 … a
1n
]. 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 = [a
ij
] with all entries a
ij
 = 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 22a
: : :
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
 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.
M01_ENGINEERING_MATHEMATICS-I _CH01_Part A.indd 2 5/30/2016 4:34:38 PM
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 I
1
, I
2
, I
3
.
In general, I
n
 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
A square matrix A = [a
ij
] is said to be an upper triangular matrix if all the entries below the main 
diagonal are zero.
That is a
ij
 = 0 if i > j
A square matrix A = [a
ij
] is said to be a lower triangular matrix if all the entries above the main 
diagonal are zero.
That is a
ij
 = 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.
M01_ENGINEERING_MATHEMATICS-I _CH01_Part A.indd 3 5/30/2016 4:34:39 PM
1.4 ■ Engineering Mathematics
1.1.1 Basic Operations on Matrices
Definition 1.10 Equality of Matrices
Two matrices A = [a
ij
] and B = [b
ij
] of the same type m × n are said to be equal if a
ij
 = b
ij
 for all i, j and 
is written as A = B.
Definition 1.11 Addition of Matrices
Let A = [a
ij
] and B = [b
ij
] of the same type m × n. Then A + B = [c
ij
], where c
ij
 = a
ij
 + b
ij
 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 = [a
ij
] be an m × n matrix and k be a scalar, then kA = [ka
ij
].
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 = [a
ij
] be an m × p matrix and B = [b
ij
] be an p × n matrix, then AB is defined and 
AB = [c
ij
] is an m × n matrix, where c a bij ik kj
k
p
=
=
∑
1
.
That is c
ij
 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 22 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
M01_ENGINEERING_MATHEMATICS-I _CH01_Part A.indd 4 5/30/2016 4:34:41 PM
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) = a A + a B (vi) (a + b)A = a A + b A
 (vii) a (bA) = (a b)A for any scalars a, b.
 2. If A, B, C are conformable for multiplication, then 
 (i) a (AB) = (a A)B = A(a B)
 (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 = [a
ij
] 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 aT 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
A square matrix A = [a
ij
] of order n is said to be symmetric if AT = A.
This means [a
ji
] = [a
ij
] ⇒ a
ji
 = a
ij
 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
A square matrix A = [a
ij
] of order n is said to be skew-symmetric if AT = −A.
This means [a
ji
] = −[a
ij
] ⇒ a
ji
 = − a
ij
 for all i, j = 1, 2, …, n
In particular, put j = i, then a
ii
 = − a
ii
 ⇒ 2a
ii
 = 0 ⇒ a
ii
 = 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.
M01_ENGINEERING_MATHEMATICS-I _CH01_Part A.indd 5 5/30/2016 4:34:41 PM
1.6 ■ Engineering Mathematics
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 Cofactorof an Element
Let A = [a
ij
] be a square matrix of order n. If we delete the row and column of the element a
ij
, we get 
a square submatrix of order (n − 1).
The determinant of this submatrix is called the minor of the element a
ij
 and is denoted by M
ij
.
The cofactor of a
ij
 in A is A Mij
i j
ij= −
+( )1
EXAMPLE 1.15
A = −
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
1 6 2
0 2 4
3 1 2
The cofactor of a
11
 = 1 is A11
1 11
2 4
1 2
= −
−+( ) = −4 −4 = −8
The cofactor of a
12
 = 6 is A12
1 21
0 4
3 2
= − +( ) = − (−12) = 12
The cofactor of a
32
 = 1 is A 32
3 21
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 = [a
ij
] 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 I
n
.
(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.
M01_ENGINEERING_MATHEMATICS-I _CH01_Part A.indd 6 5/30/2016 4:34:44 PM
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 22 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 00 2) = −
∴
 
8 12 6
10 4 17
28 4 2
8 10 28
12 4 4
6 17
=
−
− −
− −
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
=
− −
− −
−
adj A
T
22
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
∴ 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 = [a
ij
] be a complex matrix.
The conjugate matrix of A is denoted by A and A aij= ⎡⎣ ⎤⎦ .
M01_ENGINEERING_MATHEMATICS-I _CH01_Part A.indd 7 5/30/2016 4:34:46 PM
1.8 ■ Engineering Mathematics
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 aji
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 = [a
ij
], 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 a
ii
 = a + ib, then aii = a − ib
∴ a − ib = −(a + ib) ⇒ 2a = 0 ⇒ a = 0
∴ a
ii
 = 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 aA 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
M01_ENGINEERING_MATHEMATICS-I _CH01_Part A.indd 8 5/30/2016 4:34:50 PM
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 = ATA = I.
In this case A is called an orthogonal matrix. So, if A is an orthogonal matrix, then AT = A−1.
M01_ENGINEERING_MATHEMATICS-I _CH01_Part A.indd 9 5/30/2016 4:34:52 PM
1.10 ■ Engineering Mathematics
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 2i i i i i i i ii
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 ⎦⎦
⎥ = −[ ]{ i
2 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
[ *] ( *)
M01_ENGINEERING_MATHEMATICS-I _CH01_Part A.indd 10 5/30/2016 4:34:54 PM
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
2i
∴ 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.
M01_ENGINEERING_MATHEMATICS-I _CH01_Part A.indd 11 5/30/2016 4:34:57 PM
1.12 ■ Engineering Mathematics
EXAMPLE 4
If A 5
1
2 1
0 1 2
2 0
i
i1
⎡
⎣
⎢
⎤
⎦
⎥ , 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
11 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
M01_ENGINEERING_MATHEMATICS-I _CH01_Part A.indd 12 5/30/2016 4:35:00 PM
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 = I
2
.
 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 AT 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].
M01_ENGINEERING_MATHEMATICS-I _CH01_Part A.indd 13 5/30/2016 4:35:03 PM
1.14 ■ Engineering Mathematics
 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 = [a
ij
] 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 A 2
3 1
3 0
=
−⎡
⎣
⎢
⎤
⎦
⎥ and 
A 2
3 1
3 0
=
−
 is a minor of order 2. Since A 2 = 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.
M01_ENGINEERING_MATHEMATICS-I _CH01_Part A.indd 14 5/30/2016 4:35:06 PM
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 I
n
 is the unit matrix of order n, then I n = ≠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 convenientform, 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) R
i
 ↔ R
j
 means ith row and jth row are interchanged.
 (ii) R
i
 → kR
i
 means the elements of ith row is multiplied by k (≠0)
 (iii) R
i
 → R
i
 + kR
j
 means the jth row is multiplied by k and added to the ith row.
Similarly we indicate the column transformations by C
i
 ↔ C
j
, C
i
 → kC
i
, C
i
 → C
i
 + kC
j
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.
M01_ENGINEERING_MATHEMATICS-I _CH01_Part A.indd 15 5/30/2016 4:35:07 PM
1.16 ■ Engineering Mathematics
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
( )
( ) 11
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
→ + −
−
− −
−
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥ → −
( )
∼
∼
33 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
M01_ENGINEERING_MATHEMATICS-I _CH01_Part A.indd 16 5/30/2016 4:35:09 PM
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
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
, by reducing to an echelon matrix.
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
52 2 1
k
R R R( )
RR 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 RR R3 2−
M01_ENGINEERING_MATHEMATICS-I _CH01_Part A.indd 17 5/30/2016 4:35:10 PM
1.18 ■ Engineering Mathematics
 
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
− − −
−
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥ →
∼
RR R4 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
 I 3
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 R
2
 → 3R
2
 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 
I r 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 I
r
, 
I r
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 = 
I r 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 P
1
, P
2
, …, P
k
 and 
Q
1
, Q
2
, …, Q
t
 such that
 P
1
 P
2
 … P
k
 A Q
1
 Q
2
 … Q
t
 = 
I r 0
0 0
⎡
⎣
⎢
⎤
⎦
⎥
M01_ENGINEERING_MATHEMATICS-I _CH01_Part A.indd 18 5/30/2016 4:35:12 PM
Matrices ■ 1.19
⇒ PAQ = 
I r 0
0 0
⎡
⎣
⎢
⎤
⎦
⎥
where P = P
1
 P
2
 … P
k
, Q = Q
1
 Q
2
 … Q
t
Working rule to find normal form and P, Q:
Let A be a non-zero m × n matrix write A = I
m
AI
n
(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 I
m
 on R. H. S and each elementary 
column transformation of A will be applied to I
n
 on R. H. S. 
After a sequence of suitable applications of elementary transformations, we get 
I r 0
0 0
⎡
⎣
⎢
⎤
⎦
⎥ = PAQ.
Then the rank of A is the rank of I
r
 = 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
∼
∼
( )
00
0 1 0 0
0 0 1 1
1
23 3
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥ →R R
M01_ENGINEERING_MATHEMATICS-I _CH01_Part A.indd 19 5/30/2016 4:35:14 PM
1.20 ■ Engineering Mathematics
 
1 0 0 0
0 1 0 0
0 0 1 0 4 4 3
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥ → −C C C
∼
== [ : ]I 3 0
This is the normal form of A and so the r(A) = 3
EXAMPLE 2
Let A 5
2 21 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 = I
3
AI
3
⇒ 
1 1 1
1 1 1
3 1 1
1 0 0
0 1 0
0 0 1
1