Engineering Mathematics - ESE,GATE,PSUs 2019

ENGINEERING MATHEMATICS (For ESE & GATE Exam) (CE, ME, PI, CH, EC, EE, IN, CS, IT)

Salient Features : •

282 topics under 32 chapters in 8 units

•

642 Solved Examples for comprehensive understanding

•

1446 questions from last 26 years of GATE & ESE exams with detailed solutions

•

Only book having complete theory on ESE & GATE Pattern

•

Comprising conceptual questions marked with ‘*’ for quick revision

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IES MASTER PUBLICATION F-126, (Lower Basement), Katwaria Sarai, New Delhi-110016 Phone : 011-26522064, Mobile : 8130909220, 9711853908 E-mail : info@iesmasterpublications.com, info@iesmaster.org Web : iesmasterpublications.com, iesmaster.org

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Copyright Š 2018, by IES MASTER Publications. No part of this booklet may be reproduced, or distributed in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise or stored in a database or retrieval system without the prior permission of IES MASTER, New Delhi.

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First Edition : 2017 Second Edition : 2018

Typeset at : IES Master Publication, New Delhi-110016

PREFACE We, at IES MASTER, have immense pleasure in placing the second edition of “Engineering Mathematics” before the aspirants of GATE & ESE exams.

Dear Students, as we all know that in 2016 UPSC included Engineering Mathematics as a part of syllabus of common paper for ESE exam as well as of a technical paper for EC/EE branch, while Engineering Mathematics already has 15% weightage in GATE exam. We have observed that currently available books cover neither all the topics nor all previously asked questions in GATE & ESE exams. Since most of the books focus on only some selected main topics, students have not been able to answer more than 60-65% of 1446 questions that have been asked in GATE & ESE exams so far. Hence to overcome this problem, we have tried our best by covering more than 282 topics under 32 chapters in 8 units. (One should not be in dilemma that 282 topics are more than sufficient. These are the minimum topics from where GATE & ESE have already asked questions). Since we have covered every previous year questions from last 26 years of each topic, students can easily decide, how much time to allocate on each chapter based on the number of questions asked in that particular exam. Again, we have included only those proofs that are necessary for concept building of topics and we have stressed on providing elaborate solution to all the questions.

It is the only book in the market that includes complete theory exactly on ESE & GATE Pattern. After each topic there are sufficient number of solved examples for concept building & easy learning. The book includes such types of 642 examples. It also covers all the previously asked questions in which conceptual questions are marked with ‘*’ sign so that students can save their time, while revising.

Having incorporated my teaching experience of more than 14 years, I believe this book will enable the students to excel in Engineering Mathematics.

My source of inspiration is Mr. Kanchan Thakur Sir (Ex-IES). He has continuously motivated me while writing this book.

My special thanks to the entire IES MASTER Team for their continuous support in bringing out the book. I strongly believe that this book will help students in their journey to success. I invite suggestions from students, teachers & educators for further improvement in the book.

Dr. Puneet Sharma (M.Sc., Ph.D.) IES Master Publications New Delhi

CONTENTS

UNIT 1 : LINEAR ALGEBRA 1.1

Algebra of Matrices ...................................................................................... 01 – 32 (i) Definition of Matrix ........................................................................................... 01 (ii) Types of Matrices ............................................................................................. 01 (iii) Product of Matrix by a Scalar (or Constant) ....................................................... 03 (iv) Addition and Subtraction of Matrices ................................................................ 03 (v) Multiplication of Matrices .................................................................................. 03 (vi) Minors of Matrix ............................................................................................... 05 (vii) Cofactors of a Matrix ........................................................................................ 05 (viii) Properties of Determinants .............................................................................. 05 (ix) Adjoint of Matrix ............................................................................................... 06 (x) Inverse of a Matrix ............................................................................................ 06 (xi) Conjugate of a Matrix ....................................................................................... 07 (xii) Conjugate Transpose of a Matrix ...................................................................... 07 (xiii) Special Types of Matrices ................................................................................ 07 Previous Years GATE & ESE Questions ...................................................... 13

1.2

Rank of Matrices .......................................................................................... 33 – 51 (i) Definition of Rank ............................................................................................. 33 (ii) Elementary Transformations or E-Operation ..................................................... 34 (iii) Equivalent Matrices .......................................................................................... 34 (iv) Properties of Rank ........................................................................................... 34 (v) Echelon Form / Triangular Form ....................................................................... 35 (vi) Normal Form of a Matrix (Canonical Form) ....................................................... 37 (vii) Elementary Matrices ........................................................................................ 40 (viii) Computation of the Inverse of Matrix by Elementary Transformation .................. 41 (ix) Row Rank and Column Rank of a Matrix ........................................................... 44 Previous Years GATE & ESE Questions ...................................................... 46

(vi)

1.3

System of Equations ................................................................................... 52 – 92 (i) Linear Dependence and Independence of Vectors ........................................... 52 (ii) System of Linear Equations ............................................................................. 55 (iii) Methods of Solving Non-Homogenous System ................................................. 56 (iv) Methods of Solving Homogenous System ........................................................ 63 (v) LU Decomposition (Factorization) Method ....................................................... 67 (vi) Dolittle’s Method .............................................................................................. 67 (vii) Crout’s Method ................................................................................................ 70 Previous Years GATE & ESE Questions ...................................................... 73

1.4

Eigen Values and Eigen Vectors ............................................................... 93 – 138 (i) Definition ......................................................................................................... 93 (ii) Algebraic & Geometric Multiplicity .................................................................... 94 (iii) Properties of Eigen Values & Eigen Vectors .................................................... 94 (iv) Similar Matrices & Diagonalisation ................................................................ 101 (v) Cayley-Hamilton Theorem .............................................................................. 102 Previous Years GATE & ESE Questions .................................................... 106

1.5

Pivoting & Vector Space .......................................................................... 139 – 147 (i) Normalisation ................................................................................................. 139 (ii) ILL-Conditioned System ................................................................................. 139 (iii) Diagonally Dominant System ......................................................................... 139 (iv) Gauss Elimination Method with Pivoting ......................................................... 139 (v) Gauss-Jordan Method .................................................................................... 142 (vi) Vector Space ................................................................................................. 143 Previous Years GATE & ESE Questions .................................................... 145

UNIT 2 : CALCULUS 2.1

Functions and their Graphs .................................................................... 148 – 162 (i) Cartesian Product .......................................................................................... 148 (ii) Relations........................................................................................................ 148 (iii) Types of Relations .......................................................................................... 148 (iv) Functions ....................................................................................................... 149 (v) Types of Functions ......................................................................................... 149

(vii)

(vi) Some Basic Graphs....................................................................................... 150 (vii) Graph Transformation..................................................................................... 156 Previous Years GATE & ESE Questions .................................................... 157 2.2

Infinite Series ............................................................................................ 163 – 176 (i) Expansion of Functions .................................................................................. 163 (ii) Maclaurin’s Theorem (Series) ........................................................................ 163 (iii) Taylor’s Theorem (Series) .............................................................................. 163 (iv) Convergence and Divergence of Infinite Series .............................................. 166 (v) Methods to Find Convergence of Infinite Series .............................................. 167 (vi) Power Series ................................................................................................. 168 Previous Years GATE & ESE Questions .................................................... 170

2.3

Limits, Continuity and Differentiability ................................................... 177 – 220 (i) Function ......................................................................................................... 177 (ii) Limit of a Function .......................................................................................... 177 (iii) Theorem on Limits ......................................................................................... 178 (iv) Indeterminate Forms ...................................................................................... 180 (v) L-Hospital Rule .............................................................................................. 181 (vi) Fundamentals of Continuity ............................................................................ 184 (vii) Kinds of Discontinuities .................................................................................. 186 (viii) Properties of Continuous Functions ................................................................ 191 (ix) Saltus of a Function ........................................................................................ 192 (x) Function of Two Variables .............................................................................. 193 (xi) Limit of a Function of Two Variables ............................................................... 193 (xii) Continuity of Function of Two Variables .......................................................... 198 (xiii) Differentiability ............................................................................................... 199 Previous Years GATE & ESE Questions .................................................... 205

2.4

Mean Value Theorems ............................................................................. 221 – 229 (i) Rolle’s Theorem ............................................................................................. 221 (ii) Lagrange’s First Mean Value Theorem........................................................... 222 (iii) Cauchy’s Mean Value Theorem ...................................................................... 222 (iv) Bolzano Theorem ........................................................................................... 223

(viii)

(v) Intermediate Value Theorem........................................................................... 223 (vi) Darboux’s Theorem ....................................................................................... 223 Previous Years GATE & ESE Questions .................................................... 225 2.5

Maxima and Minima ................................................................................. 230 – 259 (i) Increasing-Decreasing Function ..................................................................... 230 (ii) Maxima and Minima of Function of Single Variable ......................................... 231 (iii) Maxima-Minima of Functions of Two Variables ............................................... 234 (iv) Sufficient Condition (Lagrange’s Conditions).................................................. 235 (v) Lagrange’s Method of Undetermined Multipliers ............................................. 238 Previous Years GATE & ESE Questions .................................................... 243

2.6

Partial Differentiation ............................................................................... 260 – 283 (i) Parial Derivatives ........................................................................................... 260 (ii) Homogeneous Functions ............................................................................... 265 (iii) Euler’s Theorem on Homogenous Function .................................................... 266 (iv) Total Differential Coefficient ............................................................................ 268 (v) Change of Variables ...................................................................................... 268 (vi) Jacobian ........................................................................................................ 272 (vii) Chain Rule of Jacobians ................................................................................ 275 (viii) Functional Dependence ................................................................................. 276 Previous Years GATE & ESE Questions .................................................... 279

2.7

Multiple Integral ........................................................................................ 284 – 346 (i) Concept of Integration .................................................................................... 284 (ii) Some Standard Formulae .............................................................................. 284 (iii) Some Important Integration and their Hints...................................................... 285 (iv) Definite integrals ............................................................................................ 286 (v) Fundamental Properties of Definite Integrals .................................................. 286 (vi) Fundamental Theorem of Integral Calculus ..................................................... 288 (vii) Definite integral as the Limit of a Sum ............................................................ 288 (viii) Double Integrals ............................................................................................. 121 (ix) Leibnitz Rule of Differentiation Under the Sign of Integration ........................... 291 (x) Improper Integral ............................................................................................ 292

(ix)

(xi) Beta and Gamma Functions ........................................................................... 293 (xii) Properties of Beta and Gamma Functions ...................................................... 293 (xiii) Double Integrals ............................................................................................. 300 (xiv) Double Integrals in Polar Coordinates ............................................................ 301 (xv) Triple Integrals ................................................................................................ 307 (xvi) Change of Order of Integration ....................................................................... 309 (xvii) Use of Jacobian in Multiple Integrals ............................................................... 312 (xviii) Multiple Integral using Change of Variables Concept ...................................... 314 (xix) Area and Volume in Different Coordinates Systems ....................................... 315 (xx) Arc Length of Curves (Rectification) ............................................................... 318 (xxi) Intrinsic Equation of a Curve ........................................................................... 318 (xxii) Volumes of Solids of Revolution ..................................................................... 321 (xxiii) Surfaces of Solids of Revolution ..................................................................... 322 Previous Years GATE & ESE Questions .................................................... 324

UNIT 3 : VECTOR CALCULUS 3.1

Vector & their Basic Properties ............................................................... 347 – 364 (i) Vector ............................................................................................................ 347 (ii) Dot Product.................................................................................................... 348 (iii) Cross Product ................................................................................................ 350 (iv) Scalar Triple Product ...................................................................................... 352 (v) Vector Triple Product ...................................................................................... 352 (vi) Direction Cosines & Direction Ratios ............................................................. 353 (vii) Straight Lines in 3D ........................................................................................ 354 Previous Years GATE & ESE Questions .................................................... 361

3.2

Gradient, Divergence and Curl ................................................................ 365 – 392 (i) Partial Derivatives of Vectors ......................................................................... 365 (ii) Point Functions .............................................................................................. 365 (iii) Del Operator .................................................................................................. 366 (iv) The Laplacian Operator ................................................................................. 366

(x)

(v) Gradient ......................................................................................................... 367 (vi) Directional Derivative (D.D.)........................................................................... 367 (vii) Level Surface ................................................................................................. 369 (viii) Divergence .................................................................................................... 373 (ix) Curl (Rotation) ................................................................................................ 374 (x) Vector Identities ............................................................................................. 375 Previous Years GATE & ESE Questions .................................................... 379 3.3

Vector Integration ...................................................................................... 393 – 411 (i) Line Integral ................................................................................................... 393 (ii) Surface Integral .............................................................................................. 395 (iii) Volume Integral .............................................................................................. 396 (iv) Gauss’ Divergence Theorem .......................................................................... 397 (v) Stoke’s Theorem ............................................................................................ 399 (vi) Green’s Theorem ........................................................................................... 400 Previous Years GATE & ESE Questions .................................................... 402

UNIT 4 : COMPLEX ANALYSIS 4.1

Complex Number ..................................................................................... 412 – 428 (i) IOTA .............................................................................................................. 412 (ii) Complex Numbers as Ordered Pairs.............................................................. 412 (iii) Euler Notation ................................................................................................ 413 (iv) Geometrical Representation........................................................................... 413 (v) Algebraic Properties ...................................................................................... 414 (vi) Complex Conjugate Numbers......................................................................... 415 (vii) Modulus and Argument ................................................................................... 416 (viii) Equation of a Circle in Complex Form ............................................................ 419 (ix) Complex No. as a Rotating Arrow ................................................................... 421 Previous Years GATE & ESE Questions .................................................... 422

4.2

Analytic Functions ................................................................................... 429 – 453 (i) Neighbourhood of Complex Number z0 ................................................................................... 429 (ii) Function of a Complex Variable ...................................................................... 429

(xi)

(iii) Types of Complex Function ............................................................................ 430 (iv) Complex Mapping/Transformation .................................................................. 430 (v) Continuity of Complex Function ...................................................................... 432 (vi) Differentiability of Complex Function .............................................................. 432 (vii) Analytic Functions/Regular Function/Holomorphic Function............................. 433 (viii) Cauchy-Riemann Equations ........................................................................... 434 (ix) Polar Form of Cauchy-Riemann Equations ..................................................... 436 (x) Orthogonal System......................................................................................... 438 (xi) Construction of an Analytic Function ............................................................... 442 Previous Years GATE & ESE Questions .................................................... 444 4.3

Complex Integration ................................................................................ 454 – 486 (i) Contour .......................................................................................................... 454 (ii) Elementary Properties of Complex Integrals ................................................... 455 (iii) Cauchy Integral Theorem................................................................................ 458 (iv) Cauchy’s Integral Formula .............................................................................. 459 (v) Taylor Series of Complex Function ................................................................. 464 (vi) Zeroes of an Analytic Function ........................................................................ 465 (vii) Singularities of an Analytic Function................................................................ 466 (viii) Types of Isolated Singular Points .................................................................... 466 (ix) Residue ......................................................................................................... 467 (x) Cauchy-Residue Theorem.............................................................................. 468 (xi) Methods of Evaluating Residues .................................................................... 468 Previous Years GATE & ESE Questions .................................................... 471

UNIT 5 : DIFFERENTIAL EQUATIONS 5.1

Ordinary Differential Equations............................................................... 487 – 518 (i) Definition of Differential Equations ................................................................. 487 (ii) Order and Degree of Ordinary Differential Equation ....................................... 487 (iii) Non-Linear Differential Equation .................................................................... 488 (iv) Solution of Differential Equations .................................................................... 488

(xii)

(v) Formation of Differential Equation .................................................................. 489 (vi) Wronskian...................................................................................................... 490 (vii) Linearly Dependent (LD) & Linearly Independent (LI) Solutions ....................... 490 (viii) Methods of Solving Differential Equation ........................................................ 491 (ix) Differential Equations of First Order and First Degree .................................... 491 (x) Exact Differential Equation ............................................................................. 493 (xi) Equation Reducible to Exact Form ................................................................. 494 Previous Years GATE & ESE Questions .................................................... 499 5.2

Linear Differential Equations of Higher Order with Constant Coefficients ...................................................................... 519 – 562 (i) Complementary Function (C.F.) ...................................................................... 519 (ii) Rules for Finding the Complementary Function ............................................... 519 (iii) Particular Integral (P.I.) .................................................................................... 522 (iv) Methods of Evaluating (P.I) ............................................................................. 522 (v) Cauchy’s Homogeneous Linear Differential Equation ..................................... 530 (vi) Legender’s Homogeneous Linear Differential Equations ................................ 530 (vii) Simultaneous Linear Differential Equation ...................................................... 534 (viii) Variation of Parameters ................................................................................. 534 (ix) Differential Equations of First Order and Higher Degree ................................. 535 (x) Clairaut’s Equation......................................................................................... 538 Previous Years GATE & ESE Questions .................................................... 540

5.3

Partial Differential Equations................................................................... 563 – 585 (i) Order and Degree of Partial Differential Equation ........................................... 563 (ii) Linear P.D. Equation ...................................................................................... 563 (iii) Lagrange’s Method of Solving Linear ............................................................. 564 (iv) Homogenous Linear P.D.E. with Constant Coefficient..................................... 567 (v) Non Homogeneous Linear P.D.E. with Constant Coefficient ........................... 567 (vi) Classification of 2nd Order Linear P.D.E. in Two Variables ............................. 575 (vii) One Dimensional Wave Equation ................................................................... 576 (viii) One Dimensional Heat Equation .................................................................... 577

(xiii)

(ix) Two Dimensional Heat Equation .................................................................... 578 (x) Laplace Equation ........................................................................................... 578 (xi) Clairaut Ordinary Differential Equation............................................................ 580 (xii) Clairaut Partial Differential Equation ............................................................... 580 Previous Years GATE & ESE Questions .................................................... 581

UNIT 6 : NUMERICAL METHODS 6.1

Numerical Solutions of Linear Equations .............................................. 586 – 594 (i) Gauss-Jacobi Method .................................................................................... 586 (ii) Gauss-Seidel Method of Iteration ................................................................... 589 Previous Years GATE & ESE Questions .................................................... 594

6.2

Numerical Solution of Algebraic and Transcendental Equations ........ 595 – 618 (i) Graphical Method ........................................................................................... 595 (ii) Bisection Method ........................................................................................... 596 (iii) Regula-Falsi Method ...................................................................................... 600 (iv) The Secant Method ........................................................................................ 602 (v) Newton-Raphson Method ............................................................................... 603 Previous Years GATE & ESE Questions .................................................... 607

6.3

Interpolation ............................................................................................. 619 – 627 (i) Interpolation and Extrapolation ....................................................................... 619 (ii) Various Methods of Interpolation .................................................................... 619 (iii) Notation of Finite Difference Calculus ............................................................. 619 (iv) Newton-Gregory Forward Interpolation Formula for Equal Intervals.................. 620 (v) Newton-Gregory Backward Interpolation Formula for Equal Intervals ............... 621 (vi) Newton’s Divided Difference Interpolation Formula for Unequal Intervals......... 623 (vii) Lagrange’s Interpolation Formula for Unequal Intervals ................................... 624 Previous Years GATE & ESE Questions .................................................... 626

6.4

Numerical Integration (Quadrature) ........................................................ 628 – 643 (i) General Quadrature Formula .......................................................................... 628

(xiv)

(ii) The Trapezoidal Rule ..................................................................................... 628 (iii) Simpson’s One-Third Rule ............................................................................. 629 (iv) Simpsons 3/8th Rule ....................................................................................... 630 Previous Years GATE & ESE Questions .................................................... 636 6.5

Numerical Solution of a Differential Equation ........................................ 644 – 660 (i) Picard’s Method of Successive Approximation ............................................... 644 (ii) Euler’s Method/Forward Euler Method/Explicit Euler Method .......................... 647 (iii) Backward Euler’s Method/Implict Euler Method .............................................. 647 (iv) Runge-Kutta Methods..................................................................................... 650 Previous Years GATE & ESE Questions .................................................... 656

UNIT 7 : TRANSFORM THEORY 7.1

Laplace Transform ................................................................................... 661 – 690 (i) Integral Transform........................................................................................... 661 (ii) Definition of Laplace Transform ...................................................................... 661 (iii) Properties of Laplace Transform .................................................................... 661 (iv) Laplace Transform of Periodic Function ......................................................... 462 (v) Types of Laplace Transform ........................................................................... 662 (vi) Evaluation of Integrals with the help of Laplace Transform ............................... 664 (vii) Inverse Laplace Transform ............................................................................. 665 (viii) Properties of Inverse Laplace Transform ........................................................ 665 (ix) Types of Inverse Laplace Transform................................................................ 665 (x) Convolution of Two Functions (Falting of Two Functions) ................................. 666 (xi) Unit Step Function (Heaviside’s Unit Step Function) ....................................... 666 (xii) Unit Impulse Function (Dirac Delta Function) .................................................. 666 (xiii) Existence Theorem for Laplace Transform...................................................... 668 (xiv) Initial Value Theorem ...................................................................................... 668 (xv) Final Value Theorem ...................................................................................... 668 (xvi) Application of Laplace Transform in Differential Equation ............................... 669 Previous Years GATE & ESE Questions .................................................... 671

(xv)

7.2

Fourier Series ........................................................................................... 691 – 703 (i) Periodic Function ........................................................................................... 691 (ii) Fourier Series : (Main Definition) .................................................................... 692 (iii) Euler’s Formula of Fourier Series .................................................................. 692 (iv) Dirichlet’s Conditions ..................................................................................... 694 (v) Fourier Expansion of Even and Odd Function ................................................. 695 (vi) Half Range Fourier Series .............................................................................. 697 Previous Years GATE & ESE Questions .................................................... 699

7.3

Fourier and Z-Transforms ........................................................................ 704 – 714 (i) Fourier Integral Theorem ................................................................................ 704 (ii) Fourier Transforms ......................................................................................... 705 (iii) Z-Transform (Definition) .................................................................................. 709 Previous Years GATE & ESE Questions .................................................... 713

UNIT 8 : PROBABILITY & STATISTICS 8.1

Probability ................................................................................................ 715 – 768 (i) Some Basic Concepts ................................................................................... 715 (ii) Representation of Various Events Based on Algebraic Operations ................. 718 (iii) Probability (Definition) .................................................................................... 719 (iv) AdditionTheorem of Probability ...................................................................... 723 (v) Odds in Favour and Odds Against .................................................................. 726 (vi) Independent Events ........................................................................................ 727 (vii) Conditional Probability ................................................................................... 731 (viii) Binomial Theorem on Probability .................................................................... 731 (ix) Total Probability Theorem ............................................................................... 735 (x) Baye’s Theorem (Inverse Probability Theorem) .............................................. 736 Previous Years GATE & ESE Questions .................................................... 740

8.2

Statistics - I (Probability Distributions) .................................................. 769 – 819 (i) Random Experiment ...................................................................................... 769 (ii) Discrete Random Variable ............................................................................. 769

(xvi)

(iii) The Bernoulli Random Variable ...................................................................... 771 (iv) The Binomial Random Variable ...................................................................... 771 (v) The Geometric Random Variable ................................................................... 773 (vi) The Poisson Random Variable ....................................................................... 773 (vii) Continuous Random Variable ......................................................................... 774 (viii) Uniform Random Variable .............................................................................. 776 (ix) Exponential Random Variable ........................................................................ 777 (x) Normal Random Variable/Normal Distribution/Gaussian Distribution .............. 777 (xi) Standard Normal Distribution ......................................................................... 778 (xii) Skewness in Normal Curve ............................................................................ 779 (xiii) Expectation of Random Variable .................................................................... 779 (xiv) Variance ........................................................................................................ 781 (xv) Standard Deviation ........................................................................................ 781 (xvi) Covariance .................................................................................................... 781 (xvii) Cumulative Distribution Function (CDF) for Discrete Random Variable ........... 784 (xviii) Mean ............................................................................................................. 784 (xix) Median .......................................................................................................... 784 (xx) Mode ............................................................................................................. 785 Previous Years GATE & ESE Questions .................................................... 786 8.3

Statistics - II (Correlation & Regression) ................................................ 820 – 834 (i) Curve Fitting .................................................................................................. 820 (ii) Methods of Least Suqares ............................................................................. 820 (iii) Correlation ..................................................................................................... 824 (iv) Methods of Estimating Correlation ................................................................. 825 (v) Regression Analysis....................................................................................... 727 (vi) Line of Regression of y on x ........................................................................... 827 (vii) Line of Regression of x on y ........................................................................... 828 (viii) Properties of Regression Coefficients ............................................................ 829 (ix) Angle between Two Lines of Regression ........................................................ 829 Previous Years GATE & ESE Questions .................................................... 833

1.1 DEFINITION OF MATRIX A set of m n objects or numbers (real or complex) arranged in a rectangular array of m rows and n columns, i.e.  a11 a12 a  21 a22      a  m1 am2

 a1n   a2n        amn 

is called a matrix of order m × n matrix.

The number a11, a12, ..., amn are called the elements of the matrix. The vertical lines are columns or column vectors, and the horizontal lines are called rows or row vectors of the matrix.

TYPES OF MATRICES 1. Row Matrix: A matrix having one row and any number of columns is called a row matrix, or a row vector, e.g., [a11, a12, .... a1n] or [a1, a2, ..., an] is a row matrix of order n or matrix of order 1 × n.

2. Column Matrix: A matrix having one column and any number of rows is called a column matrix or a column vector.  a11    a e.g.  21  or      am1 

 a1     a2  is a column matrix of order m or matrix of order m × 1.      am 

3. Null Matrix or Zero Matrix: Any matrix in which all the elements are zero is called a zero matrix or Null Matrix i.e. 0  0 O =  0  0

0 0 0  0 0 0 0 0 0  0 0 0

4. Square Matrix: A matrix in which the number of rows is equal to the number of columns is called a square matrix i.e. A = (aij)m×n is a square matrix if and only if m = n. A matrix  a11 a12 a a A =  21 22 a31 a32

a13  a23  a33 

is a square matrix of order 3. The elements a111, a22, a33 of the above square matrix are 33

called its diagonal elements and the diagonal containing these elements is called the principal diagonal or leading diagonal or main diagonal.

Trace of Matrix: The sum of the diagonal elements of a square matrix is called trace of the matrix.

2 Algebra of Matrices 5. Diagonal Matrix: A square matrix is called diagonal matrix if all its non-diagonal elements are zero i.e. in general a matrix A = (aij)

n×n

is called a diagonal matrix if aij = 0 for i  j; 0 0 0    A = 0 2 0  is a diagonal matrix of order 3´. 0 0 5 

For example

6. Scalar Matrix: If all the elements of a diagonal matrix of order n are equal, i.e. if aii = k  i , then the matrix is called a scalar matrix, i.e. 5 0 0    A = 0 5 0  is a scalar matrix of order 3. 0 0 5 

7. Unit or Identity Matrix: A square matrix is called a unit matrix or identity matrix if all the diagonal elements are unity and non-diagonal elements are zero. e.g.  1 0 0 1 0   I3×3 = 0 1 0  , I22   , 0 1  0 0 1

are identity matrices of order 3×3 and 2×2 respectively.

8. Upper Triangular Matrix: A square matrix A = [aij] is called an upper triangular matrix if aij = 0 whenever i > j.

Thus in an upper triangular matrix all the elements below the principal diagonal are zero. For example,

A =

1 0  0  0

2 4 2 3 1 5  and B  0 4 3  0 0 6

2 4 5  0 1 2    0 0 4 

are 4× 4 and 3×3 upper triangular matrices respectively.

9. Lower Triangular Matrix : A square matrix of A  aij  is called a lower triangular matrix if aij = 0 whenever i < j. Thus in a lower triangular matrix all the elements above the principal diagonal are zero. For example, 1  2 A 4   5

0 0 0  2 0 1 0 0   are 4×4 and 2×2 lower triangular matrices respectively.. and b   4 3 2 3 0  4 3 4 

10. Sub matrix : A matrix obtained from a given matrix, say A = (aij)m×n by deleting some rows or columns or both 1  2 is called a sub matrix of A. For example if A   7   1 of A.

2 3 4 1 2 3 3 5   3 5 1     then the matrix 2 3 5 and 8 0  are sub matrices 8 0 2 7 8 0  7 2      7 2 3 

11. Equal matrices : Two matrices are said to be equal if : (i) They are of the same order. (ii) The elements in the corresponding positions are equal. 2 3  2 3  Thus if A   , B  . Then A = B  1 4 1 4

 mn

In general if A  aij

 mn

and B  bij

are matrices each of order m×n and aij  bij for all i and j then A = B.

Engineering Mathematics

3

PRODUCT OF MATRIX BY A SCALAR (OR CONSTANT) Let A  aij  be a matrix of order m×n and k is a constant, then their product is matrix kA  kaij  i.e. every mn mn  2 1 0   8 4 0  element of A is multiplied by k. For example, if A    . Then we have 4A     4 5 3  16 20 12

ADDITION AND SUBTRACTION OF MATRICES Let A and B be two matrices of the same order, then their sum A + B is defined as the matrix each element of which is the sum of the corresponding elements of A and B.

 mn

In general if A  aij

 mn

and B  bij

then their sum is defined by the matrix.

C = A + B = (Cij)m×n

where c ij  aij  bij and i = 1, 2, ....m and j = 1, 2, 3,.......,n  1 0 2 4 2 5  If A    , B   3 1 4  11 13 6 

Then

 4 1 2  0 5  2   5 2 7  A + B =    11  3 13  1 6  4  14 14 2

Similarly, if A and B are two matrices of the same order, then their difference is defined by  4  1 2  0 5  2  3 2 3  A –B =    11  3 13  1 6  4  8 12 10 

Properties of Matrix Addition (i) Matrix addition is commutative : Let A  aij  and B  bij  be matrices of the same order m×n then A + B = B + A. mn mn (ii) Matrix addition is associative : Let A, B, C can be the matrices of the same order, Then  A  B   C  A  B  C  . (iii) Cancellation law for matrix addition : Let A, B, C be the matrices of the same order, then A + B = A + C holds if and only if B = C.

MULTIPLICATION OF MATRICES The product AB of two matrices A and B is possible only when the number of columns in A is equal to the number of rows in B. Such matrices are said to be conformable for multiplication.

Let A  aij  be a matrix of order m×n and B  b jk  be a matrix of order n×p, then the product AB is defined np mn as a matrix C  c ik mp of order m×p m

where

cik =

 aijb jk  ailblk  ai2b2k  ....  ainbnk j 1

i.e. (i, k)th element of AB = sum of the products of the elements of ith row of A with the corresponding elements of kth column of B, i.e.

IES MASTER Publication

4 Algebra of Matrices

Example 1 :

2 3    1 2 3  If A    , B   4 5  . Show that AB  BA  4 2 5   2 1  

Solution :

 1.2  2.4  3.2 1.3  2.5  3.1   0 4  AB =  4.2  2.4  5.2 4.3  2.5  5.1  10 3      22

Since number of columns in B is equal to number of rows in A, so BA is also defined. 2 3  10 2 27     1 2 3    BA =  4 5      16 2 27   4 2 5    2 1      2 2 11

Hence AB  BA .

Thus matrix multiplication is not commutative in general.

Example 2 :

0 1 0   1 2     If A  0 2 1 , B  3 0  . Find AB. Will BA exist? 2 3 1  4 1    

Solution :

0 1 0   1 2  3 0       AB = 0 2 1 3 0   10 1  2 3 1  4 1 15 5      

Now, since number of columns of B is 2 and number of rows in A are 3. So BA does not exist. Therefore if AB is defined, it is not at all necessary that the product BA is also defined.

Properties of Matrix Multiplication 1. Multiplication of matrices is not commutative i.e. AB  BA 2. Multiplication of matrices is associative, i.e. A(BC) = (AB)C 3. Matrix multiplication is distributive with respect to addition i.e. A(B + C) = AB + AC where A, B C are any three matrices of order m×n, n×p, n×p respectively.

4. Positive integral power of square matrix : The product of AA is defined only when A is square matrix of order n. We shall denote it as A2. If m and n are any positive integers, then we have A m A n  A m  n 5. Zero Divisor : AB = O does not necessarily imply that at least one of the matrices A and B must be zero matrix i.e. the product of two matrices can be zero matrix while neither of them is a zero matrix. Such matrices are said to be zero divisor.

For example; if A =

1  1

1  1 0 1 1  1 0  0 0   and B    , then AB       1  1 0  1 1  1 0  0 0 

i.e. neither the matrix A nor matrix B is a zero matrix but their product matrix AB is zero matrix.

6. Multiplication with Identity Matrix : If A be n×n matrix and In is a unit matrix of order n, then AIn  In A  A i.e. a matrix remains unaltered when it is multiplied by a unit matrix of same order.

Engineering Mathematics

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MINORS OF MATRIX The determinant value of the square matrix obtained from the original matrix of any order by the omission of the rows and columns is called a minor of a matrix. For example 1 2 3 4   If A  0 3 7 1 is a matrix of order 3 × 4. Then minors of A are 8 2 7 3    34

1 2 3 0 3 7; 8 2 7

0 3 8 2

&

7 1 7 3

etc.

COFACTORS OF A MATRIX The cofactors of the element aij is defined as Aij =

 1

i j

Mij

where |Mij| is the determinant obtained by deleting the ith row and jth column from given matrix.

EXPANSION BY COFACTORS

We have row expansion of a determinant of a general 3×3 matrix, that is a11 a12 a13

|A| = a21 a22 a 23  a11 a31 a32 a 33

Since

a22 a23 a32 a33

 M11 ,

a21 a 23 a31 a 33

 M12 , and

a 21 a 23

a 22 a23 a 32 a33

 a12

a 21 a 23 a 31 a 33

 a13

a 21 a 22 a 31 a 32

 M13 . So we have,

a31 a33

|A| = a11M11  a12M12  a13M13 = a11A 11  a12 A 12  a13 A 13 Since by the definition of cofactors, we have

and

A11 =

 111 M11  M11 ,

A12 =

 112 M12  M12 ,

A13 =

 113 M13

 M13 .

PROPERTIES OF DETERMINANTS 1. |A'| = |A| 2. If we interchange any two rows or columns then sign of determinants change.

3. If any 2 rows or any 2 columns in a determinant are identical (or proportional) then the value of the determinant is zero.

4. Multiplying a determinant by K means multiplying the elements of only one row (or one column) by K. 1 2

e.g.

then

|A| =

= –2

3 4

2| A | = 2

1

2

2 4

=

3 4

3 4

= 8 – 12 = –4 = 2(–2)

5. If elements of a row in a determinant can be expressed as the sum of two or more elements then the given determinant can be expressed as the sum of 2 or more determinants. ac bd

e.g.

|A| =

e

f

a b

=

e f



c

d

e f

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