Engineering Mathematics Kanodia

Eighth Edition

GATE

Engineering Mathematics

RK Kanodia Ashish Murolia

NODIA & COMPANY

GATE Electronics & Communication Vol 2, 8eEngineering MathematicsRK Kanodia and Ashish Murolia

Copyright By NODIA & COMPANY

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To Our Parents

Preface to the Series

For almost a decade, we have been receiving tremendous responses from GATE aspirants for our earlier books: GATE Multiple Choice Questions, GATE Guide, and the GATE Cloud series. Our first book, GATE Multiple Choice Questions (MCQ), was a compilation of objective questions and solutions for all subjects of GATE Electronics & Communication Engineering in one book. The idea behind the book was that Gate aspirants who had just completed or about to finish their last semester to achieve his or her B.E/B.Tech need only to practice answering questions to crack GATE. The solutions in the book were presented in such a manner that a student needs to know fundamental concepts to understand them. We assumed that students have learned enough of the fundamentals by his or her graduation. The book was a great success, but still there were a large ratio of aspirants who needed more preparatory materials beyond just problems and solutions. This large ratio mainly included average students.

Later, we perceived that many aspirants couldnt develop a good problem solving approach in their B.E/B.Tech. Some of them lacked the fundamentals of a subject and had difficulty understanding simple solutions. Now, we have an idea to enhance our content and present two separate books for each subject: one for theory, which contains brief theory, problem solving methods, fundamental concepts, and points-to-remember. The second book is about problems, including a vast collection of problems with descriptive and step-by-step solutions that can be understood by an average student. This was the origin of GATE Guide (the theory book) and GATE Cloud (the problem bank) series: two books for each subject. GATE Guide and GATE Cloud were published in three subjects only.

Thereafter we received an immense number of emails from our readers looking for a complete study package for all subjects and a book that combines both GATE Guide and GATE Cloud. This encouraged us to present GATE Study Package (a set of 10 books: one for each subject) for GATE Electronic and Communication Engineering. Each book in this package is adequate for the purpose of qualifying GATE for an average student. Each book contains brief theory, fundamental concepts, problem solving methodology, summary of formulae, and a solved question bank. The question bank has three exercises for each chapter: 1) Theoretical MCQs, 2) Numerical MCQs, and 3) Numerical Type Questions (based on the new GATE pattern). Solutions are presented in a descriptive and step-by-step manner, which are easy to understand for all aspirants.

We believe that each book of GATE Study Package helps a student learn fundamental concepts and develop problem solving skills for a subject, which are key essentials to crack GATE. Although we have put a vigorous effort in preparing this book, some errors may have crept in. We shall appreciate and greatly acknowledge all constructive comments, criticisms, and suggestions from the users of this book. You may write to us at [email protected] and [email protected].

Acknowledgements

We would like to express our sincere thanks to all the co-authors, editors, and reviewers for their efforts in making this project successful. We would also like to thank Team NODIA for providing professional support for this project through all phases of its development. At last, we express our gratitude to God and our Family for providing moral support and motivation.

We wish you good luck ! R. K. KanodiaAshish Murolia

SYLLABUS

Engineering Mathematics (EC, EE, and IN Branch )

Linear Algebra: Matrix Algebra, Systems of linear equations, Eigen values and eigen vectors.

Calculus: Mean value theorems, Theorems of integral calculus, Evaluation of definite and improper integrals, Partial Derivatives, Maxima and minima, Multiple integrals, Fourier series. Vector identities, Directional derivatives, Line, Surface and Volume integrals, Stokes, Gauss and Greens theorems.

Differential equations: First order equation (linear and nonlinear), Higher order linear differential equations with constant coefficients, Method of variation of parameters, Cauchys and Eulers equations, Initial and boundary value problems, Partial Differential Equations and variable separable method.

Complex variables: Analytic functions, Cauchys integral theorem and integral formula, Taylors and Laurent series, Residue theorem, solution integrals.

Probability and Statistics: Sampling theorems, Conditional probability, Mean, median, mode and standard deviation, Random variables, Discrete and continuous distributions, Poisson, Normal and Binomial distribution, Correlation and regression analysis.

Numerical Methods: Solutions of non-linear algebraic equations, single and multi-step methods for differential equations.

Transform Theory: Fourier transform, Laplace transform, Z-transform.

Engineering Mathematics (ME, CE and PI Branch )

Linear Algebra: Matrix algebra, Systems of linear equations, Eigen values and eigen vectors. Calculus: Functions of single variable, Limit, continuity and differentiability, Mean value theorems, Evaluation of definite and improper integrals, Partial derivatives, Total derivative, Maxima and minima, Gradient, Divergence and Curl, Vector identities, Directional derivatives, Line, Surface and Volume integrals, Stokes, Gauss and Greens theorems.

Differential equations: First order equations (linear and nonlinear), Higher order linear differential equations with constant coefficients, Cauchys and Eulers equations, Initial and boundary value problems, Laplace transforms, Solutions of one dimensional heat and wave equations and Laplace equation.

Complex variables: Analytic functions, Cauchys integral theorem, Taylor and Laurent series.

Probability and Statistics: Definitions of probability and sampling theorems, Conditional probability, Mean, median, mode and standard deviation, Random variables, Poisson,Normal and Binomial distributions.

Numerical Methods: Numerical solutions of linear and non-linear algebraic equations Integration by trapezoidal and Simpsons rule, single and multi-step methods for differential equations.

********** .

CONTENTS

CHAPTER 1 MATRIX ALGEBRA

1.1 INTRODUCTION 1

1.2 MULTIPLICATION OF MATRICES 1

1.3 TRANSPOSE OF A MATRIX 2

1.4 DETERMINANT OF A MATRIX 2

1.5 RANK OF MATRIX 2

1.6 ADJOINT OF A MATRIX 3

1.7 INVERSE OF A MATRIX 3

1.7.1 Elementary Transformations 4

1.7.2 Inverse of Matrix by Elementary Transformations 4

1.8 ECHELON FORM 5

1.9 NORMAL FORM 5

EXERCISE 6

SOLUTIONS 20

CHAPTER 2 SYSTEMS OF LINEAR EQUATIONS

2.1 INTRODUCTION 39

2.2 VECTOR 39

2.2.1 Equality of Vectors 39

2.2.2 Null Vector or Zero Vector 39

2.2.3 A Vector as a Linear Combination of a Set of Vectors 40

2.2.4 Linear Dependence and Independence of Vectors 40

2.3 SYSTEM OF LINEAR EQUATIONS 40

2.4 SOLUTION OF A SYSTEM OF LINEAR EQUATIONS 40

EXERCISE 42

SOLUTIONS 51

CHAPTER 3 EIGENVALUES AND EIGENVECTORS

3.1 INTRODUCTION 65

3.2 EIGENVALUES AND EIGEN VECTOR 65

3.3 DETERMINATION OF EIGENVALUES AND EIGENVECTORS 66

3.4 CAYLEY-HAMILTON THEOREM 66

3.4.1 Computation of the Inverse Using Cayley-Hamilton Theorem 67

3.5 REDUCTION OF A MATRIX TO DIAGONAL FORM 67

3.6 SIMILARITY OF MATRICES 68

EXERCISE 69

SOLUTIONS 80

CHAPTER 4 LIMIT, CONTINUITY AND DIFFERENTIABILITY

4.1 INTRODUCTION 99

4.2 LIMIT OF A FUNCTION 99

4.2.1 Left Handed Limit 99

4.2.2 Right Handed Limit 99

4.2.3 Existence of Limit at Point 100

4.2.4 L hospitals Rule 100

4.3 CONTINUITY OF A FUNCTION 100

4.3.1 Continuity in an interval 100

4.4 DIFFERENTIABILITY 101

EXERCISE 102

SOLUTIONS 115

CHAPTER 5 MAXIMA AND MINIMA

5.1 INTRODUCTION 139

5.2 MONOTONOCITY 139

5.3 MAXIMA AND MINIMA 139

EXERCISE 140

SOLUTIONS 147

CHAPTER 6 MEAN VALUE THEOREM


6.2 ROLLES THEOREM 163

6.3 LAGRANGES MEAN VALUE THEOREM 163

6.4 CAUCHYS MEAN VALUES THEOREM 163

EXERCISE 164

SOLUTIONS 168

CHAPTER 7 PARTIAL DERIVATIVES


7.2 PARTIAL DERIVATIVES 175

7.2.1 Partial Derivatives of Higher Orders 175

7.3 TOTAL DIFFERENTIATION 176

7.4 CHANGE OF VARIABLES 176

7.5 DIFFERENTIATION OF IMPLICIT FUNCTION 176

7.6 EULERS THEOREM 176

EXERCISE 177

SOLUTIONS 182

CHAPTER 8 DEFINITE INTEGRAL


8.2 DEFINITE INTEGRAL 191

8.3 IMPORTANT FORMULA FOR DEFINITE INTEGRAL 192

8.4 DOUBLE INTEGRAL 192

EXERCISE 193

SOLUTIONS 202

CHAPTER 9 DIRECTIONAL DERIVATIVES


9.2 DIFFERENTIAL ELEMENTS IN COORDINATE SYSTEMS 223

9.3 DIFFERENTIAL CALCULUS 223

9.4 GRADIENT OF A SCALAR 224

9.5 DIVERGENCE OF A VECTOR 224

9.6 CURL OF A VECTOR 225

9.7 CHARACTERIZATION OF A VECTOR FIELD 225

9.8 LAPLACIAN OPERATOR 225

9.9 INTEGRAL THEOREMS 226

9.9.1 Divergence theorem 226

9.9.2 Stokes Theorem 226

9.9.3 Greens Theorem 226

9.9.4 Helmholtzs Theorem 226

EXERCISE 227

SOLUTIONS 234

CHAPTER 10 FIRST ORDER DIFFERENTIAL EQUATIONS


10.2 DIFFERENTIAL EQUATION 247

10.2.1 Ordinary Differential Equation 247

10.2.2 Order of a Differential Equation 248

10.2.3 Degree of a Differential Equation 248

10.3 DIFFERENTIAL EQUATION OF FIRST ORDER AND FIRST DEGREE 248

10.4 SOLUTION OF A DIFFERENTIAL EQUATION 249

10.5 VARIABLES SEPARABLE FORM 249

10.5.1 Equations Reducible to Variable Separable Form 250

10.6 HOMOGENEOUS EQUATIONS 250

10.6.1 Equations Reducible to Homogeneous Form 251

10.7 LINEAR DIFFERENTIAL EQUATION 252

10.7.1 Equations Reducible to Linear Form 253

10.8 BERNOULLIS EQUATION 253

10.9 EXACT DIFFERENTIAL EQUATION 254

10.9.1 Necessary and Sufficient Condition for Exactness 254

10.9.2 Solution of an Exact Differential Equation 254

10.9.3 Equations Reducible to Exact Form: Integrating Factors 255

10.9.4 Integrating Factors Obtained by Inspection 255

EXERCISE 257

SOLUTIONS 266

CHAPTER 11 HIGHER ORDER DIFFERENTIAL EQUATIONS


11.2 LINEAR DIFFERENTIAL EQUATION 283

11.2.1 Operator 283

11.2.2 General Solution of Linear Differential Equation 283

11.3 DETERMINATION OF COMPLEMENTARY FUNCTION 284

11.4 PARTICULAR INTEGRAL 285

11.4.1 Determination of Particular Integral 285

11.5 HOMOGENEOUS LINEAR DIFFERENTIAL EQUATION 287

11.6 EULER EQUATION 288

EXERCISE 289

SOLUTIONS 300

CHAPTER 12 INITIAL AND BOUNDARY VALUE PROBLEMS


12.2 INITIAL VALUE PROBLEMS 317

12.3 BOUNDARY-VALUE PROBLEM 317

EXERCISE 319

SOLUTIONS 325

CHAPTER 13 PARTIAL DIFFERENTIAL EQUATION


13.2 PARTIAL DIFFERENTIAL EQUATION 337

13.2.1 Partial Derivatives of First Order 337

13.2.2 Partial Derivatives of Higher Order 338

13.3 HOMOGENEOUS FUNCTIONS 339

13.4 EULERS THEOREM 339

13.5 COMPOSITE FUNCTIONS 340

13.6 ERRORS AND APPROXIMATIONS 341

EXERCISE 342

SOLUTIONS 347

CHAPTER 14 ANALYTIC FUNCTIONS


14.2 BASIC TERMINOLOGIES IN COMPLEX FUNCTION 357

14.3 FUNCTIONS OF COMPLEX VARIABLE 358

14.4 LIMIT OF A COMPLEX FUNCTION 358

14.5 CONTINUITY OF A COMPLEX FUNCTION 359

14.6 DIFFERENTIABILITY OF A COMPLEX FUNCTION 359

14.6.1 Cauchy-Riemann Equation: Necessary Condition for Differentiability of a Complex Function 360

14.6.2 Sufficient Condition for Differentiability of a Complex Function 361

14.7 ANALYTIC FUNCTION 362

14.7.1 Required Condition for a Function to be Analytic 362

14.8 HARMONIC FUNCTION 363

14.8.1 Methods for Determining Harmonic Conjugate 363

14.8.2 Milne-Thomson Method 364

14.8.3 Exact Differential Method 366

14.9 SINGULAR POINTS 366

EXERCISE 367

SOLUTIONS 380

CHAPTER 15 CAUCHYS INTEGRAL THEOREM


15.2 LINE INTEGRAL OF A COMPLEX FUNCTION 405

15.2.1 Evaluation of the Line Integrals 405

15.3 CAUCHYS THEOREM 406

15.3.1 Cauchys Theorem for Multiple Connected Region 407

15.4 CAUCHYS INTEGRAL FORMULA 408

15.4.1 Cauchys Integral Formula for Derivatives 409

EXERCISE 410

SOLUTIONS 420

CHAPTER 16 TAYLORS AND LAURENT SERIES


16.2 TAYLORS SERIES 439

16.3 MACLAURINS SERIES 440

16.4 LAURENTS SERIES 441

16.5 RESIDUES 443

16.5.1 The Residue Theorem 443

16.5.2 Evaluation of Definite Integral 443

EXERCISE 444

SOLUTIONS 453

CHAPTER 17 PROBABILITY


17.2 SAMPLE SPACE 469

17.3 EVENT 469

17.3.1 Algebra of Events 470

17.3.2 Types of Events 470

17.4 DEFINITION OF PROBABILITY 471

17.4.1 Classical Definition 471

17.4.2 Statistical Definition 472

17.4.3 Axiomatic Definition 472

17.5 PROPERTIES OF PROBABILITY 472

17.5.1 Addition Theorem for Probability 472

17.5.2 Conditional Probability 473

17.5.3 Multiplication Theorem for Probability 473

17.5.4 Odds for an Event 473

17.6 BAYES THEOREM 474

EXERCISE 476

SOLUTIONS 491

CHAPTER 18 RANDOM VARIABLE


18.2 RANDOM VARIABLE 515

18.2.1 Discrete Random Variable 516

18.2.2 Continuous Random Variable 516

18.3 EXPECTED VALUE 517

18.3.1 Expectation Theorems 517

18.4 MOMENTS OF RANDOM VARIABLES AND VARIANCE 518

18.4.1 Moments about the Origin 518

18.4.2 Central Moments 518

18.4.3 Variance 518

18.5 BINOMIAL DISTRIBUTION 519

18.5.1 Mean of the Binomial distribution 519

18.5.2 Variance of the Binomial distribution 519

18.5.3 Fitting of Binomial Distribution 520

18.6 POISSON DISTRIBUTION 521

18.6.1 Mean of Poisson Distribution 521

18.6.2 Variance of Poisson Distribution 521

18.6.3 Fitting of Poisson Distribution 522

18.7 NORMAL DISTRIBUTION 522

18.7.1 Mean and Variance of Normal Distribution 523

EXERCISE 526

SOLUTIONS 533

CHAPTER 19 STATISTICS


19.2 MEAN 543

19.3 MEDIAN 544

19.4 MODE 545

19.5 MEAN DEVIATION 545

19.6 VARIANCE AND STANDARD DEVIATION 546

EXERCISE 547

SOLUTIONS 550

CHAPTER 20 CORRELATION AND REGRESSION ANALYSIS


20.2 CORRELATION 555

20.3 MEASURE OF CORRELATION 555

20.3.1 Scatter or Dot Diagrams 555

20.3.2 Karl Pearsons Coefficient of Correlation 556

20.3.3 Computation of Correlation Coefficient 557

20.4 RANK CORRELATION 558

20.5 REGRESSION 559

20.5.1 Lines of Regression 559

20.5.2 Angle between Two Lines of Regression 560

EXERCISE 562

SOLUTIONS 565

CHAPTER 21 SOLUTIONS OF NON-LINEAR ALGEBRAIC EQUATIONS


21.2 SUCCESSIVE BISECTION METHOD 569

21.3 FALSE POSITION METHOD (REGULA-FALSI METHOD) 569

21.4 NEWTON - RAPHSON METHOD (TANGENT METHOD) 570

EXERCISE 21 571

SOLUTIONS 21 579

CHAPTER 22 INTEGRATION BY TRAPEZOIDAL AND SIMPSONS RULE


22.2 NUMERICAL DIFFERENTIATION 597

22.2.1 Numerical Differentiation Using Newtons Forward Formula 597

22.2.2 Numerical Differentiation Using Newtons Backward Formula 598

22.2.3 Numerical Differentiation Using Central Difference Formula 599

22.3 MAXIMA AND MINIMA OF A TABULATED FUNCTION 599

22.4 NUMERICAL INTEGRATION 600

22.4.1 Newton-Cotes Quadrature Formula 600

22.4.2 Trapezoidal Rule 601

22.4.3 Simpsons One-third Rule 601

22.4.4 Simpsons Three-Eighth Rule 602

EXERCISE 604

SOLUTIONS 608

CHAPTER 23 SINGLE AND MULTI STEP METHODS FOR DIFFERENTIAL EQUATIONS


23.2 PICARDS METHOD 617

23.3 EULERS METHOD 618

23.3.1 Modified Eulers Method 618

23.4 RUNGE-KUTTA METHODS 619

23.4.1 Runge-Kutta First Order Method 619

23.4.2 Runge-Kutta Second Order Method 619

23.4.3 Runge-Kutta Third Order Method 619

23.4.4 Runge-Kutta Fourth Order Method 620

23.5 MILNES PREDICTOR AND CORRECTOR METHOD 620

23.6 TAYLORS SERIES METHOD 621

EXERCISE 623

SOLUTIONS 628

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Matrix Algebra

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GATE Engineering Mathematics EC, EE, ME, CE, PI, IN

Sample Chapter of Engineering Mathematics

1.1 INTRODUCTION

This chapter, concerned with the matrix algebra, includes the following topics: Multiplication of matrix Transpose of matrix Determinant of matrix Rank of matrix Adjoint of matrix Inverse of matrix: elementary transformation, determination of inverse

using elementary transformation

Echelon form and normal form of matrix; procedure for reduction of normal form.

1.2 MULTIPLICATION OF MATRICES

If A and B be any two matrices, then their product AB will be defined only when number of columns in A is equal to the number of rows in B .If A aij m n= #6 @and B bjk n p= #6 @then their product,

AB cC ik m p= = #6 @will be matrix of order m p# , where

cik a bijj

n

jk1

==/

PROPERTIES OF MATRIX MULTIPLICATION

If A, B and C are three matrices such that their product is defined, then1. Generally not commutative; AB BA!

2. Associative law; ( ) ( )AB C A BC=3. Distributive law; ( )A B C AB AC+ = +4. Cancellation law is not applicable, i.e. if AB AC= , it does not mean

that B C= .5. If AB 0= , it does not mean that A 0= or B 0= .6. ( ) ( )AB BAT T=

CHAPTER 1MATRIX ALGEBRA

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1.3 TRANSPOSE OF A MATRIX

The matrix obtained form a given matrix A by changing its rows into columns or columns int rows is called Transpose of matrix A and is denoted by AT . From the definition it is obvious that if order of A is m n# , then order of AT is n m# .

PROPERTIES OF TRANSPOSE OF MATRIX

Consider the two matrices A and B1. ( )A AT T =2. ( )A B A BT T T! !=3. ( )AB B AT T T=4. ( ) ( )k kA AT T=5. ( ... ) ...A A A A A A A A A An n T nT nT T T T1 2 3 1 1 3 2 1=- -

1.4 DETERMINANT OF A MATRIX

The determinant of square matrix A is defined as

A aaa

aaa

aaa

11

21

31

12

22

32

13

23

33

=

PROPERTIES OF DETERMINANT OF MATRIX

Consider the two matrices A and B .1. A exists A+ is a square matrix2. AB A B=3. A AT =4. ,k kA An= if A is a square matrix of order n .5. If A and B are square matrices of same order then AB BA= .6. If A is a skew symmetric matrix of odd order then A 0= .7. If A = diag ( , ,...., )a a an1 2 then , ...a a aA n1 2= .8. ,n NA An n != .9. If A 0= , then matrix is called singular.

Singular Matrix

A square matrix A is said to be singular if A 0= and non-singular if .A 0!

1.5 RANK OF MATRIX

The number, r with the following two properties is called the rank of the matrix1. There is at least one non-zero minor of order r .

2. Every minor of order ( )r 1+ is zero.This definition of the rank does uniquely fix the same for, as a consequence

of the condition (2), every minor of order ( ),r 2+ being the sum of multiples of minors of order ( ),r 1+ will be zero. In fact, every minor of order greater

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than r will be zero as a consequence of the condition (2).The given following two simple results follow immediately from the definition 1. There exists a non-zero minor or order r & the rank is r$ .

2. All minors of order ( )r 1+ are zero & the rank is r# .In each of the two cases above, we assume r to satisfy only one of two

properties (1) or (2) of the rank. The rank of matrix A represented by symbol ( ) .Ar

Nullity of a Matrix

If A is a square matrix of a order ,n then ( )n Ar- is called the nullity of the matrix A and is denoted by ( )N A . Thus a non-singular square matrix of order n has rank equal to n and the nullity of such a matrix is equal to zero.

1.6 ADJOINT OF A MATRIX

If every element of a square matrix A be replaced by its cofactor in A , then the transpose of the matrix so obtained is called the Adjoint of matrix A and it is denoted by adj A. Thus, if aA ij= 6 @ be a square matrix and Fij be the cofactor of aij in A , then adj FA ij T+ 6 @ .

PROPERTIES OF ADJOINT MATRIX

If A, B are square matrices of order n ad In is corresponding unit matrix, then1. A (adj A) IA n= = (adj A)A2. A Aadj n 1= -3. adj (adj A) A An 2= -4. A Aadj(adj ) ( )n 1

2= -5. ) ( )A Aadj( adjT T=6. AB B Aadj( ) (adj )(adj )=7. ) ( ) ,m NA Aadj( adjm m !=8. ( ) ( ),k k k RA Aadj adjn 1 != -

1.7 INVERSE OF A MATRIX

If A and B are two matrices such that AB I BA= = , then B is called the inverse of A and it is denoted by A 1- . Thus, A 1- B AB I BA+= = =To find inverse matrix of a given matrix A we use following formula

A 1- AAadj=

Thus A 1- exists if A 0! and matrix A is called invertible.



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PROPERTIES OF INVERSE MATRIX

Let A and B are two invertible matrices of the same order, then1. ( ) ( )A AT T1 1=- -2. AB B A( ) 1 1 1=- - -3. ( ) ( ) ,k NA Ak k1 1 !=- -4. A Aadj( ) (adj )1 1=- -5. A A A

11 1= =- -

6. If A = diag ( , ,..., )a a an1 2 , then A 1=- diag ( , , ..., )a a an1 1 2 1 1- - -7. AB AC B C&= = , if A 0!

1.7.1 Elementary Transformations

Any one of the following operations on a matrix is called an elementary transformation (or E -operation).

1. Interchange of two rows or two columns

(1) The interchange of i th and j th rows is denoted by R Ri j)

(2) The interchange of i th and j th columns is denoted by C Ci j) .

2. Multiplication of (each element ) a row or column by a .k

(1) The multiplication of i th row by k is denoted by R kRi i"

(2) The multiplication of i th column by k is denoted by C kCi i"

3. Addition of k times the elements of a row (or column) to the corresponding elements of another row (or column), k 0!

(1) The addition of k times the j th row to the i th row is denoted by R R kRi i j" + .

(2) The addition of k times the j th column to the i th column is denoted by C C kCi i j" + .If a matrix B is obtained from a matrix A by one or more E -operations,

then B is said to be equivalent to A. They can be written as A B+ .

1.7.2 Inverse of Matrix by Elementary Transformations

The elementary row transformations which reduces a square matrix A to the unit matrix, when applied to the unit matrix, gives the inverse matrix A 1- . Let A be a non-singular square matrix. Then, A IA=Apply suitable E -row operations to A on the left hand side so that A is reduced to I . Simultaneously, apply the same E -row operations to the pre-factor I on right hand side. Let I reduce to ,B so that I BA= . Post-multiplying by A 1- , we get IA 1- BAA 1= - or A 1- ( )B AA BI B1= = =-or B A 1= -

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Sample Chapter of Engineering Mathematics1.8 ECHELON FORM

A matrix is said to be of Echelon form if,1. Every row of matrix A which has all its entries 0 occurs below every row

which has a non-zero entry.

2. The first non-zero entry in each non-zero is equal to one.

3. The number of zeros before the first non-zero element in a row is less than the number of such zeros in the next row.

Rank of a matrix in the Echelon form

The rank of a matrix in the echelon form is equal to the number of non-zero rows of the given matrix. For example,

the rank of the matrix A 000

200

610

120

3 4

=#

R

T

SSSS

V

X

WWWW

is 2

1.9 NORMAL FORM

By a finite number of elementary transformations, every non-zero matrix A of order m n# and rank ( )r 0> can be reduced to one of the following forms.

, , ,IO

OO

IO I O I

r rr r> > 8 8H H B B

Ir denotes identity matrix of order r . Each one of these four forms is called Normal Form or Canonical Form or Orthogonal Form.

Procedure for Reduction of Normal Form

Let aA ij= 6 @ be any matrix of order m n# . Then, we can reduce it to the normal form of the matrix A by subjecting it to a number of elementary transformation using following methodology.

METHODOLOGY: REDUCTION OF NORMAL FORM

1. We first interchange a pair of rows (or columns), if necessary, to obtain a non-zero element in the first row and first column of the matrix A.

2. Divide the first row by this non-zero element, if it is not 1.

3. We subtract appropriate multiples of the elements of the first row from other rows so as to obtain zeroes in the remainder of the first column.

4. We subtract appropriate multiple of the elements of the first column from other columns so as to obtain zeroes in the remainder of the first row.

5. We repeat the above four steps starting with the element in the second row and the second column.

6. Continue this process down the leading diagonal until the end of the diagonal is reached or until all the remaining elements in the matrix are zero.

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EXERCISE 1

QUE 1.1 Match the items in column I and II.

Column I Column II

P. Singular matrix 1. Determinant is not defined

Q. Non-square matrix 2. Determinant is always one

R. Real symmetric 3. Determinant is zero

S. Orthogonal matrix 4. Eigenvalues are always real

5. Eigenvalues are not defined

(A) P-3, Q-1, R-4, S-2

(B) P-2, Q-3, R-4, S-1

(C) P-3, Q-2, R-5, S-4

(D) P-3, Q-4, R-2, S-1

QUE 1.2 Cayley-Hamilton Theorem states that a square matrix satisfies its own characteristic equation. Consider a matrix

A32

20=

--> H

If A be a non-zero square matrix of orders n , then(A) the matrix A A+ l is anti-symmetric, but the matrix A A- l is

symmetric

(B) the matrix A A+ l is symmetric, but the matrix A A- l is anti-symmetric

(C) Both A A+ l and A A- l are symmetric(D) Both A A+ l and A A- l are anti-symmetric

QUE 1.3 If A and B are two odd order skew-symmetric matrices such that AB BA=, then what is the matrix AB ?(A) An orthogonal matrix

(B) A skew-symmetric matrix

(C) A symmetric matrix

(D) An identity matrix

QUE 1.4 If A and B are matrices of order 4 4# such that A B5= and A Ba =, then a is_______.

ARIHANT/286/26

ARIHANT/285/3

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Sample Chapter of Engineering MathematicsQUE 1.5 If the rank of a 5 6#^ h matrix A is 4, then which one of the following

statements is correct?(A) A will have four linearly independent rows and four linearly independent

columns

(B) A will have four linearly independent rows and five linearly independent columns

(C) AAT will be invertible(D) A AT will be invertible

QUE 1.6 If An n# is a triangular matrix then det A is

(A) ( )a1 iii

n

1

-=% (B) aii

i

n

1=%

(C) ( )a1 iii

n

1

-=/ (D) aii

i

n

1=/

QUE 1.7 If ,RA n n! # det A 0! , then A is(A) non singular and the rows and columns of A are linearly independent.(B) non singular and the rows A are linearly dependent.(C) non singular and the A has one zero rows.(D) singular

QUE 1.8 Square matrix A of order n over R has rank n . Which one of the following statement is not correct?(A) AT has rank n(B) A has n linearly independent columns(C) A is non-singular(D) A is singular

QUE 1.9 Determinant of the matrix 513

325

2610

R

T

SSSS

V

X

WWWW is_____

QUE 1.10 The value of the determinant

ahg

hbf

gfc

(A) abc fgh af bg ch2 2 2 2+ - - - (B) ab a c d+ + +(C) abc ab bc cg+ - - (D) a b c+ +

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ARIHANT/286/28

ARIHANT/305/7



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QUE 1.11 The value of the determinant 673981

191324

211426

is______

QUE 1.12 If 102

357

268

26- = , then the determinant of the matrix 201

753

862

-R

T

SSSS

V

X

WWWW is____

QUE 1.13 The determinant of the matrix

0101

1102

0100

2311

-

-

R

T

SSSSS

V

X

WWWWW

is______

QUE 1.14 Let A be an m n# matrix and B an n m# matrix. It is given that determinant I ABm+ =^ h determinant I BAn+^ h, where Ik is the k k# identity matrix. Using the above property, the determinant of the matrix given below is______2111

1211

1121

1112

R

T

SSSSSS

V

X

WWWWWW

QUE 1.15 Let ii

A3

1 21 2

2= --> H, then

(1) ii

A3

1 21 2

2= +-> H (2) * i iA 21 2 1 22= - +> H

(3) *A A= (4) A is hermitian matrixWhich of above statement is/are correct ?(A) 1 and 3 (B) 1, 2 and 3

(C) 1 and 4 (D) All are correct

QUE 1.16 For which value of l will the matrix given below become singular?

8412

06

020

l R

T

SSSS

V

X

WWWW

QUE 1.17 If 012

102

23l

--

-R

T

SSSS

V

X

WWWW is a singular, then l is______

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Sample Chapter of Engineering MathematicsQUE 1.18 Multiplication of matrices E and F is G . matrices E and G are

cossin

sincosE

0 0

001

qq

qq=

-R

T

SSSS

V

X

WWWW and G

100

010

001

=R

T

SSSS

V

X

WWWW

What is the matrix F ?

(A) cossin

sincos

0 0

001

qq

qq

-R

T

SSSS

V

X

WWWW (B)

coscos

cossin

0 0

001

qq

qq-

R

T

SSSS

V

X

WWWW

(C) cossin

sincos

0 0

001

qq

qq-

R

T

SSSS

V

X

WWWW (D)

sincos

cossin

0 0

001

qq

qq

-R

T

SSSS

V

X

WWWW

QUE 1.19 Rank of matrix 12

20

35

47-= G is

QUE 1.20 The rank of the matrix 111

111

101

-R

T

SSSS

V

X

WWWW is______

QUE 1.21 Given,

A 112

246

325

=R

T

SSSS

V

X

WWWW

(1) A 0= (2) A 0=Y(3) rank A 2=^ h (4) rank A 3=^ hWhich of above statement is/are correct ?(A) 1, 3 and 4 (B) 1 and 3

(C) 1, 2 and 4 (D) 2 and 4

QUE 1.22 Given, A202

134

123

=---

R

T

SSSS

V

X

WWWW is

(1) A 0= (3) A 0=Y(3) rank A 2=^ h (4) rank A 5=^ hWhich of above statement is/are correct ?(A) 1, 3 and 4 (B) 1 and 3

(C) 1, 2 and 4 (D) 2 and 4

QUE 1.23 Given matrix A462

231

140

371

=R

T

SSSS

6V

X

WWWW

@ , the rank of the matrix is_____

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QUE 1.24 The rank of the matrix A241

174

3

5l =

-R

T

SSSS

V

X

WWWW is 2. The value of l must be

QUE 1.25 The rank of matrix

1236

2428

3317

0235

R

T

SSSSS

V

X

WWWWW

is______

QUE 1.26 Given,

A

acac

bdbd

1010

0101

=

(1) A 0= (2) Two rows are identical(3) rank A 2=^ h (4) rank A 3=^ hWhich of above statement is/are correct ?(A) 1, 3 and 4 (B) 1 and 3

(C) 1, 2 and 3 (D) 2 and 4

QUE 1.27 Two matrices A and B are given below :

A pr

qs= > H; p qpr qs

pr qsr s

B2 2

2 2= ++++> H

If the rank of matrix A is N , then the rank of matrix B is(A) /N 2 (B) N 1-(C) N (D) N2

QUE 1.28 If , ,x y z are in AP with common difference d and the rank of the matrix

k

xyz

456

56

R

T

SSSS

V

X

WWWW is 2, then the value of d and k are

(A) /d x 2= ; k is an arbitrary number (B) d an arbitrary number; k 7=(C) d k= ; k 5= (D) /d x 2= ; k 6=

QUE 1.29 The rank of a 3 3# matrix C AB= , found by multiplying a non-zero column matrix A of size 3 1# and a non-zero row matrix B of size 1 3# , is(A) 0 (B) 1

(C) 2 (D) 3

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QUE 1.30 If xxx

yyy

A111

1

2

3

1

2

3

=R

T

SSSS

V

X

WWWW and the point ( , ),( , ),( , )x y y y x y1 1 2 2 3 2 are collinear, then the

rank of matrix A is(A) less than 3 (B) 3

(C) 1 (D) 0

QUE 1.31 Let [ ], ,a i j nA 1ij # #= with n 3$ and .a i jij = Then the rank of A is(A) 0 (B) 1

(C) n 1- (D) n

QUE 1.32 Let P be a matrix of order m n# , and Q be a matrix of order ,n p n p# ! . If ( ) nPr = and ( ) pQr = , then rank (PQ)r is(A) n (B) p

(C) np (D) n p+

QUE 1.33 x x xx n1 2 Tg= 8 B is an n-tuple nonzero vector. The n n# matrixV xxT=(A) has rank zero (B) has rank 1

(C) is orthogonal (D) has rank n

QUE 1.34 If , ,x y z in A.P. with common difference d and the rank of the matrix

k

xyz

456

56

R

T

SSSS

V

X

WWWW is 2, then the values of d and k are respectively

(A) x4

and 7 (B) 7, and x4

(C) x7

and 5 (D) 5, and x7

QUE 1.35 If the rank of a ( )5 6# matrix Q is 4, then which one of the following statement is correct ?(A) Q will have four linearly independent rows and four linearly independent

columns

(B) Q will have four linearly independent rows and five linearly independent columns

(C) QQT will be invertible

(D) Q QT will be invertible



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QUE 1.36 The adjoint matrix of 10

42

-= G is(A)

40

21= G (B) 14 02= G

(C) 21

40= G (D) 20 41= G

QUE 1.37 If xz

ybA = > H, then adj(adj A) is equal to

(A) by

zx-

-= G (B) by zx= G(C)

xb yzby

yx

1- -= G (D) None of these

QUE 1.38 If A is a 3 3# matrix and A 2= then A (adj A) is equal to

(A) 400

040

004

R

T

SSSS

V

X

WWWW (B)

200

020

002

R

T

SSSS

V

X

WWWW

(C) 100

010

001

R

T

SSSS

V

X

WWWW (D) 0

0

0

0

00

21

21

21

R

T

SSSS

V

X

WWWW

QUE 1.39 If A is a 2 2# non-singular square matrix, then adj(adj A) is(A) A2 (B) A(C) A 1- (D) None of the above

Common Data For Q. 40 to 42If A is a 3 - rowed square matrix such that A 3= .

QUE 1.40 The adj(adj A) is equal to(A) 3A (B) 9A(C) 27A (D) 81A

QUE 1.41 The value of Aadj(ajd ) is equal to(A) 3 (B) 9

(C) 27 (D) 81

QUE 1.42 The value of Aadj(adj )2 is equal to(A) 34 (B) 38

(C) 316 (D) 332

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Sample Chapter of Engineering MathematicsQUE 1.43 The rank of an n row square matrix A is ( )n 1- , then

(A) adjA 0! (B) adjA 0=(C) adj IA n= (D) adj IA n 1= -

QUE 1.44 The adjoint of matrix A122

212

221

=- -

-

--

R

T

SSSS

V

X

WWWW is equal to

(A) A (B) 3A(C) 3AT (D) At

QUE 1.45 The matrix, that has an inverse is

(A) 36

12= G (B) 52 21= G

(C) 69

23= G (D) 84 21= G

QUE 1.46 The inverse of the matrix A13

25=

--> H is

(A) 53

21= G (B) 53 31-= G

(C) 53

21

--

--= G (D) 52 31= G

QUE 1.47 The inverse of the 2 2# matrix 15

27= G is

(A) 31 7

521

--= G (B) 31

75

21= G

(C) 31 7

521-

-= G (D) 3175

21

--

--= G

QUE 1.48 If B is an invertible matrix whose inverse in the matrix 35

46= G, then B is

(A) 65

46-

-= G (B) 5 431

61> H

(C) 3 2

25

23

--= G (D) 3

1

51

41

61> H

QUE 1.49 Matrix AC

BM 0= > H is an orthogonal matrix. The value of B is

(A) 21 (B)

21

(C) 1 (D) 0



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QUE 1.50 If A121

211

=R

T

SSSS

V

X

WWWW then A 1- is

(A) 232

317

R

T

SSSS

V

X

WWWW (B)

121

212

--R

T

SSSS

V

X

WWWW

(C) 132

425

R

T

SSSS

V

X

WWWW (D) Undefined

QUE 1.51 The inverse of matrix A153

021

002

=R

T

SSSS

V

X

WWWW is equal to

(A) 41

251

021

002

-- -

R

T

SSSS

V

X

WWWW (B)

21

251

011

002

-- -

R

T

SSSS

V

X

WWWW

(C) 1101

021

002

-- -

R

T

SSSS

V

X

WWWW (D)

41

4101

021

002

-- -

R

T

SSSS

V

X

WWWW

QUE 1.52 If det A = 7, where adg

beb

cfc

A =R

T

SSSS

V

X

WWWW then det( )A2 1- is_____

QUE 1.53 If R122

013

112

=--

R

T

SSSS

V

X

WWWW, the top of R 1- is

(A) [ , , ]5 6 4 (B) [ , , ]5 3 1-(C) [ , , ]2 0 1- (D) [ , , ]2 1 21-

QUE 1.54 Let B be an invertible matrix and inverse of 7B is 14

27

--= G, the matrix B

is

(A) 1

74

72

71> H (B) 74 21= G

(C) 1

72

74

71> H (D) 72 41= G

QUE 1.55 If A

0001

1234

0021

0100

=

R

T

SSSSSS

V

X

WWWWWW

, then det A( )1- is equal to_____

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QUE 1.56 Given an orthogonal matrix A

1110

1110

1101

1101

= -- -

-

R

T

SSSSSS

V

X

WWWWWW

, AAT 1-6 @ is

(A) 000

0

00

00

0

000

41

41

21

21

R

T

SSSSSS

V

X

WWWWWW

(B) 000

0

00

00

0

000

21

21

21

21

R

T

SSSSSS

V

X

WWWWWW

(C)

1000

0100

0010

0001

R

T

SSSSS

V

X

WWWWW

(D) 000

0

00

00

0

000

41

41

41

41

R

T

SSSSSS

V

X

WWWWWW

QUE 1.57 Given an orthogonal matrix

A

1110

1110

1101

1101

= -- -

R

T

SSSSS

V

X

WWWWW

AAT 1-6 @ is

(A) 000

0

00

00

0

000

41

41

21

21

R

T

SSSSSS

V

X

WWWWWW

(B) 000

0

00

00

0

000

21

21

21

21

R

T

SSSSSS

V

X

WWWWWW

(C)

1000

0100

0010

0001

R

T

SSSSS

V

X

WWWWW

(D) 000

0

00

00

0

000

41

41

41

41

R

T

SSSSSS

V

X

WWWWWW

QUE 1.58 A is m n# full rank matrix with m n> and I is identity matrix. Let matrix ( )A A A AT T1= -l , Then, which one of the following statement is FALSE ?

(A) AA A A=l (B) ( )AA 2l(C) IA A =l (D) AA A A=l l

QUE 1.59 For a matrix xM53

54

53=6 >@ H, the transpose of the matrix is equal to the

inverse of the matrix, M MT 1= -6 6@ @ . The value of x is given by(A) 5

4- (B) 53-

(C) 53 (D) 5

4

QUE 1.60 If xx xA2 0= > H and A 11

02

1

-- > H then the value of x is_____



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QUE 1.61 The value of 23

12

21

-= =G G is(A)

83= G (B) 38= G

(C) ,3 8- -8 B (D) [ , ]3 8

QUE 1.62 If A13

21

04= -> H, then AAT is

(A) 11

34-= G (B) 11 02 13-= G

(C) 51

126= G (D) Undefined

QUE 1.63 If A23

15

11

720

- = --> >H H, then the matrix A is equal to(A)

13

25= G (B) 25 13= G

(C) 52

31= G (D) 52 31-= G

QUE 1.64 Let, .

A20

0 13=

-> H and abA 01 21

=- > H. Then ( )a b+ =_____

QUE 1.65 Let .

A20

0 13=

-> H and abA 01 21

=- > H, Then ( )a b+ is(A)

207 (B)

203

(C) 2019 (D)

2011

QUE 1.66 If A23

69= > H and y

xB

32= > H, then in order that AB 0= , the values of x and

y will be respectively(A) 6- and 1- (B) 6 and 1(C) 6 and 3- (D) 5 and 14

QUE 1.67 If A11

10

01= > H and B

101

=R

T

SSSS

V

X

WWWW, the product of A and B is

(A) 10= G (B) 10 01= G

(C) 12= G (D) 10 02= G

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QUE 1.68 If A12

11

20= > H and B

121

201

=-

R

T

SSSS

V

X

WWWW, then ( )AB T is

(A) 14

44= G (B) 11 44= G

(C) 14

41= G (D) 14 14= G

QUE 1.69 If A213

104

=-

-R

T

SSSS

V

X

WWWW and B

13

24

50=

- -> H then AB is

(A) 119

8222

10515

--

--

-R

T

SSSS

V

X

WWWW (B)

010

0221

10515

- ----

R

T

SSSS

V

X

WWWW

(C) 119

8222

10515

- --

--

R

T

SSSS

V

X

WWWW (D)

019

8221

10515

--

--

R

T

SSSS

V

X

WWWW

QUE 1.70 If X

0000

1000

0100

0010

=

R

T

SSSSSS

V

X

WWWWWW

, then the rank of X XT , where XT denotes the transpose

of X , is______

QUE 1.71 Consider the matrices ,X Y( ) ( )4 3 4 3# # and P( )2 3# . The order of [ ( ) ]P X Y PT T T T

will be(A) ( )2 2# (B) ( )3 3#

(C) ( )4 3# (D) ( )3 4#

QUE 1.72 If cos

sinsincosA

a aa= -a > H, then consider the following statements :

1. .A A A=a b ab 2. .A A A( )=a b a b+3. ( )

cossin

sincosA

nn

n

n

n

aa

aa= -a > H 4. ( ) cossin sincosnn nnA n aa aa= -a > H

Which of the above statements are true ?

(A) 1 and 2 (B) 2 and 3

(C) 2 and 4 (D) 3 and 4

QUE 1.73 If tantan

A0

022= -aa> H then ( ) cossin sincosI A 2aa a- -

a > H is equal to(A) I A2- (B) I A-(C) I A2+ (D) I A+

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QUE 1.74 Let cossin

sincosA

qq

qq= -> H

(1) AA10

01

T = > H (2) AA 1T =(3) A is orthogonal matrix (4) A is not a orthogonal matrix

Which of above statement is/are correct ?(A) 1, 3 and 4

(B) 2 and 3

(C) 1, 2 and 3

(D) 2 and 4

QUE 1.75 If the product of matrices

A cos

cos sincos sin

sin

2

2q

q qq q

q= = G and B

coscos sin

cos sinsin

2

2f

f ff f

f= = Gis a null matrix, then q and f differ by(A) an even multiple 2

p (B) an even multiple p(C) an odd multiple of 2

p (D) an odd multiple of p

QUE 1.76 For a given 2 2# matrix A, it is observed that A11

11- =- -> >H H and

A12 2

12- =- -> >H H. The matrix A is

(A) A21

11

10

02

11

12= - -

-- - -> > >H H H

(B) A11

12

10

02

21

11= - - - -> > >H H H

(C) A11

12

10

02

21

11= - -

-- - -> > >H H H

(D) 01

23

--= G

QUE 1.77 If A31

41=

--> H, then for every positive integer ,n An is equal to

(A) n

nn

n1 2 4

1 2+

+= G (B) nn nn1 2 41 2- -+= G(C)

nn

nn

1 2 41 2

-+= G (D) nn nn1 2 41 2+ --= G

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QUE 1.78 For which values of the constants b and c is the vector bc

3R

T

SSSS

V

X

WWWW a linear

Combination of ,132

264

R

T

SSSS

R

T

SSSS

V

X

WWWW

V

X

WWWW and

132

---

R

T

SSSS

V

X

WWWW

(A) 9, 6 (B) 6, 9

(C) 6, 6 (D) 9, 9

QUE 1.79 The values of non zero numbers , , , , , , ,a b c d e f g h such that the matrix adf

bkg

ceh

R

T

SSSS

V

X

WWWW

is invertible for all real numbers k .

(A) finite soln (B) infinite soln

(C) 0 (D) none

***********



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SOLUTIONS 1

SOL 1.1 Correct option is (A).(P) Singular Matrix " Determinant is zero A 0=(Q) Non-square matrix " An m n# matrix for which m n! , is called non- square matrix. Its determinant is not defined(R) Real Symmetric Matrix " Eigen values are always real.(S) Orthogonal Matrix " A square matrix A is said to be orthogonal if AA IT =Its determinant is always one.

SOL 1.2 Correct option is (B).Here, if A be a non-zero square matrix of order n , then the matrix A A+ l is symmetric, but A A- l will be anti-symmetric.

SOL 1.3 Correct option is (C).If A and B are both order skew-symmetric matrices, then A AT=- and B BT=- ...(1)Also, given that AB BA= B AT T= - -^ ^h h [from Eq. (1)] B AT T= AB T= ^ h AB AB T= ^ h i.e., AB is a symmetric matrix

SOL 1.4 Correct answer is 625.If k is a constant and A is a square matrix of order n n# then k kA An= . A B5= & A B B B5 5 6254= = =or a 625=

SOL 1.5 Correct option is (A).If rank of 5 6#^ h matrix is 4, then surely it must have exactly 4 linearly independent rows as well as 4 linearly independent columns.Hence, Rank = Row rank = Column rank

SOL 1.6 Correct option is (B).From linear algebra for An n# triangular matrix det A is equal to the product of the diagonal entries of A.

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Sample Chapter of Engineering MathematicsSOL 1.7 Correct option is (B).

If det ,A 0! then An n# is non-singular, but if An n# is non-singular, then no row can be expressed as a linear combination of any other. Otherwise det A = 0

SOL 1.8 Correct option is (D).Since, if A is a square matrix of rank n , then it cannot be a singular.

SOL 1.9 Correct answer is 28- .

513

325

2610

5(20 30) 3(10 18) 2(5 6)= +- - - -

50 24 2 28=- + - =-

SOL 1.10 Correct option is (A).

ahg

hbf

gfc

aaf

fc h

hg

fc g

hg

bf= - +

a bc f h hc fg g hf gb2= - - - + -^ ^ ^h h h abc af h c hfg ghf g b2 2 2= = - + + - abc fgh af bg ch2 2 2 2= + - - -

SOL 1.11 Correct answer is 43- . Determinant 67

1324

1426 19

3981

1426 21

3981

1324= - +

134 2280 2457= + - 43=-

SOL 1.12 Correct answer is 26.By interchanging any row or column, the value of determinant will remain same. For the given matrix, the first and third row are interchanged, thus the value remains the same.

SOL 1.13 Correct answer is 1- .

We have A

0101

1102

0100

2311

= -

-

R

T

SSSSS

V

X

WWWWW

Expanding cofactor of a34

A 1011

112

010

=- --

[ ( ) ]0 1 0 1 0=- - - + 1=-



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SOL 1.14 Correct answer is 5.Consider the given matrix be

I ABm+ 2111

1211

1121

1112

=

R

T

SSSSSS

V

X

WWWWWW

where m 4= so, we obtain

AB

2111

1211

1121

1112

1000

0100

0010

0001

= -

R

T

SSSSSS

R

T

SSSSSS

V

X

WWWWWW

V

X

WWWWWW

1111

1111

1111

1111

=

R

T

SSSSSS

V

X

WWWWWW

1111

=

R

T

SSSSSS

V

X

WWWWWW

1 1 1 16 @

Hence, we get A

1111

=

R

T

SSSSSS

V

X

WWWWWW

, B 1 1 1 1= 6 @

Therefore, BA 1 1 1 1

1111

=

R

T

SSSSSS

8V

X

WWWWWW

B 4= 6 @From the given property,

Det I ABm+^ h Det I BAn= +^ h

Det

2111

1211

1121

1112

R

T

SSSSSS

V

X

WWWWWW

Det 1 4= +6 6@ @" ,

Det 5= 6 @ 5=NOTE : Determinant of identity matrix is always 1.

SOL 1.15 Correct option is (D).

A = conjugate of A i

i31 2

1 22= ++> H

and A A* T= =^ h transpose of A i

i31 2

1 22= -+> H

Since, A* A=Hence, A is hermitian matrix.

SOL 1.16 Correct answer is 4.For singularity of matrix,

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8412

06

020

l 0=

8 0 12 0 2 12 0#l - - - =^ ^h h 4& l =

SOL 1.17 Correct answer is 2- .Matrix A is singular if A 0=

012

102

23l

--

-R

T

SSSS

V

X

WWWW 0=

or ( )112

22

10

23 0

02

3l l- - -- + - + - 0=

or ( ) ( )4 2 3l- + 0=or l 2=-

SOL 1.18 Correct option is (C).

Given EF G= where G I= = Identity matrix

cossin

sincos F

0 0

001#

qq

qq

-R

T

SSSS

V

X

WWWW

0 01

00

10

01

=R

T

SSSS

V

X

WWWW

We know that the multiplication of a matrix and its inverse be a identity matrix

AA 1- I=So, we can say that F is the inverse matrix of E

F E 1= - .adj EE=6 @

adjE ( )cos

sinsincos

0 0

001

Tqq

qq=

-R

T

SSSS

V

X

WWWW

cossin

sincos

0 0

001

qq

qq= -

R

T

SSSS

V

X

WWWW

E ( )cos cos sin sin0 0 0# #q q q q= - - - - +^ ^h h6 8@ B cos sin 12 2q q= + =

Hence, F .adjEE= 6 @ cossin sincos

0 0

001

qq

qq= -

R

T

SSSS

V

X

WWWW

SOL 1.19 Correct answer is 2.

We have A 12

20

35

47= -= G

It is a 2 4# matrix, thus )A(r 2#The second order minor

12

20- 4 0!=

Hence, ( )Ar 2=



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SOL 1.20 Correct answer is 2.We have

A 111

111

101

110

110

100

+= - -R

T

SSSS

R

T

SSSS

V

X

WWWW

V

X

WWWW R R3 1-

Since one full row is zero, ( )A 3

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Thus A 241

174

3

50l=

-=

or ( ) ( ) ( )235 4 1 20 3 16 7l l- + - + - 0=or 70 8 20 27l l- + - + 0 0= =or, l 13=

SOL 1.25 Correct answer is 3.It is 4 4# matrix, So its rank ( )A 4#r

We have A

1236

2428

3317

0235

=

1230

2420

3310

0230

= applying ( )R R R R R4 1 2 3 4"- + +

1000

2040

3380

0230

= --- applying

R R RR R R

23

2 1 2

3 1 3

"

"

--

The only fourth order minor is zero.

Since the third order minor ( )( )( )100

240

383

1 4 3 12 0!- --

= - - =

Therefore its rank is ( )A 3r =

SOL 1.26 Correct option is (C).Here, A 0=All minors of order 3 are zero, since two rows are identical.

The second minor 10

01 0=Y

Hence, Rank A^ h 2=


Given the two matrices,

A pr

qs= > H

and B p qpr qs

pr qsr s

2 2

2 2= ++++> H

To determine the rank of matrix A, we obtain its equivalent matrix using

the operation, a i2 a aa ai i2

11

211! - as



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A p q

s pr q0= -> H

If s pr q- 0=

or ps rq- 0=then rank of matrix A is 1, otherwise the rank is 2. Now, we have the matrix

B p qpr qs

pr qsr s

2 2

2 1= ++++> H

To determine the rank of matrix B , we obtain its equivalent matrix using

the operation, a i2 a aa ai i2

11

211! - as

B p q pr qs

r sp qpr qs

0

2 2

2 22 2

2=+ +

+ - ++^ ^h h

R

T

SSSS

V

X

WWWW

If r sp qpr qs2 2

2 2

2

+ - ++^ ^h h ps rq 2= -^ h 0=

or ps rq- 0=then rank of matrix B is 1, otherwise the rank is 2.Thus, from the above results, we conclude that

If ps rq 0- = , then rank of matrix A and B is 1.If ps rq 0!- , then rank of A and B is 2.i.e. the rank of two matrices is always same. If rank of A is N then rank of B also N .

SOL 1.28 Correct option is (B).It is given that , ,x y z are in A.P. with common difference d

x x= , y x d= + , z x d2= +

Let A k

xyz

456

56=

k

xx dx d

456

56

2= +

+

k

xdd

411

51

6=

-Applying R R R2 1 2- = and R R R3 2 3- =

k

xd

410

51

7 0=

- R R R3 3 2= -

A 0= k d x7 4 0& - - =^ ^h h d x4= , k 7= .

SOL 1.29 Correct option is (B).

Let A abc

1

1

1

=R

T

SSSS

V

X

WWWW, a b cB 2 2 2= 6 @

Let C AB=

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abc

a b ca ab ac a

a bb bc b

a cb cc c

1

1

1

2 2 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

#= =R

T

SSSS

R

T

SSSS

8V

X

WWWW

V

X

WWWW

BThe 3 3# minor of this matrix is zero and all the 2 2# minors are also zero. So the rank of this matrix is 1, i.e.

Cr 6 @ 1=

SOL 1.30 Correct option is (A).Since all point are collinear,

Thus xxx

yyy

111

1

2

3

1

2

3

0=

Therefore A( )r 3 >H H 10 01= = G

or x

x20

02= G 10 01= = G

So, x x2 121&= =


23

12

21

-> >H H ( )( ) ( )( )( )( ) ( )( )2 2 1 13 2 2 1=

+ -+> H 38= = G


AAT 13

21

04

120

314

= - -R

T

SSSS

>V

X

WWWW

H

( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )

( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( )

1 1 2 2 0 03 1 1 2 4 0

1 3 2 1 0 43 3 1 1 4 4=

+ ++ - +

+ - ++ - - +> H

51

126= > H


A 11

720

23

15

1

= --- -> >H H

11

720 13

1 53

12=

-- - -

--b l> >H H

131 1

1720

53

12= - -

--> >H H

131 26

651339= -

--> H

25

13= = G

SOL 1.64 Correct answer is 0.35.

We have A .2

00 13=

-= G and abA 01 21

=- > HNow AA 1- I=

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or . a

b20

0 13 0

21-> >H H 10 01= > H

or .a b

b10

2 0 13-> H 10

01= > H

or .a2 0 1- 0= and b3 1=Thus solving above we have b

31= and a

601=

Therefore a b+ 31

601

207= + =


We know that AA 1- I=or

. ab

20

0 13 0

21-> >H H .a bb10 2 0 13 10 01= - => >H H

We get b3 1= or b31=

and .a b2 0 1- 0= or a b20

=

Thus a b+ 31

31

201

207= + =


We have A 23

69= = G and ,y xB AB3 2 0= => H

We get yx2

369

32= =G G 00 00= = G

or yy

xx

6 69 9

2 123 18

++

++= G 00 00= = G

We get y6 6+ 0= or y 1=-and x2 12+ 0= or x 6=-


AB 11

10

01

101

=R

T

SSSS

=V

X

WWWW

G

( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( )1 1 1 0 0 11 1 0 0 1 1

12

+ +

+ +== = =G G


We have AB 12

11

20

121

201

=-

R

T

SSSS

=V

X

WWWW

G 14 44= = G

( )AB T 14

44= = G



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AB 213

104

13

24

50= -

- - -R

T

SSSS

=V

X

WWWW

G

( )( ) ( )( )( )( ) ( )( )

( )( ) ( )( )

( )( ) ( )( )( )( ) ( )( )

( )( ) ( )( )

( )( ) ( )( )( )( ) ( )( )

( )( ) ( )( )

2 1 1 31 1 0 33 1 4 3

2 2 1 41 2 0 43 2 4 4

2 5 1 01 5 0 03 5 4 0

=+ -+

- +

- + -- +

- - +

- + -- +

- - +

R

T

SSSS

V

X

WWWW

119

8222

10515

=- -

---

R

T

SSSS

V

X

WWWW

SOL 1.70 Correct answer is 3.

We have X

0000

1000

0100

0010

=

R

T

SSSSSS

V

X

WWWWWW

and transpose of X , XT

0100

0010

0001

0000

=

R

T

SSSSSS

V

X

WWWWWW

Here, we can see that rank of matrix x 3= , hence, we can determine the rank of X XT .Let Y X XT$= , the rank of Y# rank of X . Also, X Y XT1 =- and so we have

Rank X = Rank XT # Rank of YHence, from Eqs. (1) and (2), we get

Rank of X = Rank of YHence, rank of X XT$ is 3


X4 3# XcT

4" #

X YT3 4 4 3# # ( )X YT

3 3" #

( )X YY 3 3# ( )X YT 1

3 3" #-

P2 3# PT3 2" #

( )X Y PT T3 31 3 2# #- ( )X YT PT1 3 2" #

-" , {( ) }P X Y PT T2 3 1 3 2# #

- ( )P X Y PT T1 2 2" #-6 @

[ ( ) ]P X Y PT T T1 2 2#- [ ( ) ]P X Y PT T1 2 2" #

-


.A Aa b cossin

sincos

cossin

sincos

aa

aa

bb

bb= - -> >H H

( )( )

( )( )

cossin

sincos A

a ba b

a ba b=

+- +

++ = a b+> H

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Also, it is easy to prove by induction that

( )A na cossin

sincos

nn

nn

aa

aa= -= G


Let tan t2a =

Then, cosa tantan

t tt

11 1

22

22

2

2= +- = +

-aa

and sina tantan

tt

12

12

22

22= + = +a

a

( )cossin

sincosI A

aa

aa-

-> H tantan cos

sinsincos

112

2#

aa

aa= -

-a

a> =H G

( )

( )t

t tt

tt

tt

tt

11

11

12

12

11

2

2

2

2

2

2#= -+-

+

+-

+-

R

T

SSSSS

=V

X

WWWWW

G

(tantan

tt

I A1

11

1 )22= - = - = +aa> >H H


AAT cossin

sincos

cossin

sincos

qq

qq

qq

qq= -

-> >H H

cos sinsin cos cos sin

cos sin sin cossin cos

2 2

2 2

q qq q q q

q q q qq q=

+- +

- ++> H

10

01= > H

1= , Hence A is orthogonal matrix.


AB ( )( )

( )( )

cos cos coscos sin cos

cos sin cossin sin cos

q f q ff a a f

a f q fq f q f=

--

--= G

Is a null matrix when ( )cos 0q f- = , this happens when ( )q f- is an odd multiple of

2p .


Let matrix A be ac

bd= G

From A11

11- =- -> >H H we get ac bd a bc d11 11- = -- =- -= = = =G G G G

We have a b- 1=- ...(1)and c d- 1= ...(2)



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From A12 2

12- =- -> >H H we get

ac

bd= G 12-= G a bc d22 2 12= -- =- -= =G G

we have a b2- 2=- ...(2) c d2- 4= ...(4)Solving equation (1) and (3) a 0= and b 1=Solving equation (2) and (4) c 2=- and d 3=-Thus A

02

13= - -= G

If we check all option then result of option C after multiplication gives result.


A2 31

41

31

41

52

83=

--

-- =

--= = =G G G

If we put n 2= in option, then only D satisfy.


bc

3R

T

SSSS

V

X

WWWW k k k

132

264

132

1 2 3= + +---

R

T

SSSS

R

T

SSSS

R

T

SSSS

V

X

WWWW

V

X

WWWW

V

X

WWWW

k k k132

2132

132

1 2 3+ -R

T

SSSS

R

T

SSSS

R

T

SSSS

V

X

WWWW

V

X

WWWW

V

X

WWWW b

c

3=

R

T

SSSS

V

X

WWWW

( )k k k2132

1 2 3+ -R

T

SSSS

V

X

WWWW b

c

3=

R

T

SSSS

V

X

WWWW

k k k21 2 3+ - 3= k k k3 6 31 2 3+ - b= k k k2 6 21 2 3+ - c=& b 9= , c 6=


( )det A ( )ah cf k bef cdg aeg bdh= - + + - -Thus matrix A is invertible for all k if (and only if) the coefficient ( )ah cf- of k is 0, while the sum bef cdg aeg bdh+ - - is non zero.& Thus infinitely many other soln

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