HIGHER MATHEMATICS FOR ENGINEERING AND TECHNOLOGY Problems and Solutions www.Technicalbookspdf.com www.Technicalbookspdf.com HIGHER MATHEMATICS FOR ENGINEERING AND TECHNOLOGY Problems and Solutions Mahir M Sabzaliev, PhD Ilhama M Sabzalieva, PhD www.Technicalbookspdf.com Apple Academic Press Inc Apple Academic Press Inc 3333 Mistwell Crescent Spinnaker Way Oakville, ON L6L 0A2 Waretown, NJ 08758 Canada USA ©2018 by Apple Academic Press, Inc No claim to original U.S Government works Printed in the United States of America on acid-free paper International Standard Book Number-13: 978-1-77188-642-0 (Hardcover) International Standard Book Number-13: 978-0-203-73013-3 (eBook) All rights reserved No part of this work may be reprinted or reproduced or utilized in any form or by any electronic, mechanical or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publisher or its distributor, except in the case of brief excerpts or quotations for use in reviews or critical articles This book contains information obtained from authentic and highly regarded sources Reprinted material is quoted with permission and sources are indicated Copyright for individual articles remains with the authors as indicated A wide variety of references are listed Reasonable efforts have been made to publish reliable data and information, but the authors, editors, and the publisher cannot assume responsibility for the validity of all materials or the consequences of their use The authors, editors, and the publisher have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained If any copyright material has not been acknowledged, please write and let us know so we may rectify in any future reprint Trademark Notice: Registered trademark of products or corporate names are used only for explanation and identification without intent to infringe Library and Archives Canada Cataloguing in Publication Sabzaliev, Mahir M., author Higher mathematics for engineering and technology : problems and solutions / Mahir M Sabzaliev, PhD, Ilhama M Sabzalieva, PhD Includes bibliographical references and index Issued in print and electronic formats ISBN 978-1-77188-642-0 (hardcover). ISBN 978-0-203-73013-3 (PDF) Engineering mathematics I Sabzaliev, Ilhama M., author II Title TA330.S23 2018 620.001’51 C2018-900583-1 C2018-900584-X CIP data on file with US Library of C ongress Apple Academic Press also publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronic format For information about Apple Academic Press products, visit our website at www.appleacademicpress.com and the CRC Press website at www.crcpress.com www.Technicalbookspdf.com ABOUT THE AUTHORS Mahir M Sabzaliev Mahir M Sabzaliev is head of Higher Mathematics and Technical Sciences chair at Baku Business University He is also a professor of general and applied mathematics at Azerbaijan University of Oil and Industry, Baku, Azerbaijan, where he was a head of the higher mathematics chair in 2011– 2015 He is a member of the International Teachers Training Academy of Science He has authored over 100 published scientific works, including 30 educational works and scientific-methodical aids He has given many talks at international conferences His papers were published in several wellknown journals, including Doklady Academy of Sciences of SSSR, Doklady of Russian Academy of Sciences, Differential Equations (Differentsial’nye Uravneniya), and Uspekhi Matematicheskikh Nauk, among others Dr Sabzaliev graduated from Azerbaijan State Pedagogical University with an honors diploma in mathematics He worked as a teacher of mathematics in a secondary school and subsequently enrolled as a full-time post graduate student and earned the candidate of physical-mathematical sciences degree In 2013, he earned a PhD in mathematics Ilhama M Sabzalieva Ilhama M Sabzalieva, PhD, is an associate professor of general and applied mathematics and also department chair at Azerbaijan University of Oil and Industry, Baku, Azerbaijan She has authored over 40 scientific works, including 10 educational works and scientific-methodical aids She has authored more than 40 scientific works and has prepared educational supplies and scientific-methodical aids She has attended several international conferences and given talks She has also published papers in journals such as Doklady of Russian Academy of Sciences and Differentsial’nye Uravneniya Dr Sabzalieva graduated from Azerbaijan State University of Oil and Industry with an honors diploma and also earned the candidate of physical-mathematical sciences degree www.Technicalbookspdf.com www.Technicalbookspdf.com CONTENTS Preface ix Elements of Linear Algebra and Analytic Geometry Introduction to Mathematical Analysis 115 Differential Calculus of a Function of One Variable 183 Studying Functions of Differential Calculus and Their Application to Construction of Graphs 221 Higher Algebra Elements 243 Index 257 www.Technicalbookspdf.com www.Technicalbookspdf.com PREFACE This book was prepared and written based on the long-term pedagogical experience of the authors at Azerbaijan State University of Oil and Industry The problems that cover all themes of mathematics on engineeringtechnical specialties of higher technical schools have been gathered in this volume, which consists of three parts The volume contains sections on “elements of linear algebra and analytic geometry,” “differential calculus of a function of one variable,” and “higher algebra elements.” In this book, on every theme we present short theoretical materials and then give problems to be solved in class or independently at home, along with their answers On each theme we give the solution of some typical, relatively difficult problems and guidelines for solving them In the case of when students will be working out the problems to be solved independently, we have taken into account the problems’ similarity with the problems to be solved in class, and we stress the development of the self-dependent thinking ability of students The problems marked by “*” are relatively difficult and are intended for students who want to work independently This book is intended for bachelor students of engineering-technical specialties of schools of higher education and will also be a good resource for those beginning in various engineering and technical fields The book will also be valuable to mathematics faculty, holders of master’s degrees, engineering staff, and others www.Technicalbookspdf.com 248 Higher Mathematics for Engineering and Technology Answer: 1) 3 = 3(cos 0° + i sin 0°; 2) −1 = 1(cos π + i sin π); π π π π 3) 3i = cos + i sin ; 4) − 2i = cos − + i sin − ; 2 5) − 2i = 2 (cos (−45°) + i sin (−45°); 7) cos 5π 5π + i sin 6 Problem 442 Calculate: 1) (1 − i 3)6 ; 2) (−1 + i )5 ; 3) (− + i )6 Answer: 1) 64; 2) 4(1−i); 3) –64 Guideline: Write the numbers in trigonometric form, and use (5.6) Problem 443 Find: 1) 1; 2) −1; 3) i ; 4) −1 + i Solution of 1): Write the number under the radical sign, and use (5.7): 0° + 360° ⋅ k 0° + 360° ⋅ k + i sin = 3 = cos (120°k ) + i sin (120°k ); k = 0,1, = cos ° + i sin 0° = cos k = 0 for k = 1 for ( 1) k = 2 for ( 1) Answer: 2) ± i, ± 4) 3 ( 1) = cos 0° + i sin 0° = 1, = cos120° + i sin120° = − + i , 2 = cos 240° + i sin 240° = − − i 2 1 ± i ; 3) ± + i ; 2 2 (cos (45° + 120° ⋅ k ) + i sin (45° + 120° ⋅ k )); k = 0,1, Problem 444 Solve the equation: x4 + = Answer: + i, − i, − + i , − − i Higher Algebra Elements 249 Home tasks Problem 445 Execute the operations: 1) (a + bi ) (a − bi ); 2) (3 − 2i ) ; 3) 1+ i + 4i ; 4) 1− i 2i Answer: 1) a + b ; 2) − 12i ; 3) i ; 4) − i Problem 446 Decompose polynomials into linear co-factors: 1) x + 1; 2) x − 1; 3) x − x + Answer: 1) ( x + i )( x − i ); 2) ( x + i )( x − i )( x + 1)( x − 1); 3) ( x − + 2i )( x − − 2i ) Problem 447 Write the complex numbers in trigonometric form: 1) z = 3; 2) z = −2; 3) z = − i ; 4) z = 5i ; 5) z = + i ; 6) z = −1 + i ; 7) z = − i ; 8) z = −1 − i Answer: 1) ( cos 0° + i sin 0° ) ; 2) ( cos π + i sin π ) ; 3) 1 π π π π cos − + i sin − ; 4) cos + i sin ; 2 2 2 5) π π cos + i sin ; 6) 4 3π 3π cos + i sin 4 7) π π cos − + i sin − ; 8) ; 3π cos − 3π + i sin − Problem 448 Calculate: 1) i ; 2) −2 + 2i ; 3) −1 + i ; 4) −8 + 3i Answer: 1) ± + i, − i ; 2) + i, ( cos165° + i sin165° ) , ( cos 285° + i sin 285° ) ; 3) 2(cos ϕ + i sin ϕ ), ϕ = 45°,165°, 285°; 4) + i, − + 3i, − − i, − 3i 250 Higher Mathematics for Engineering and Technology Problem 449 Solve the equation: x3 + = Answer: −2, ± i 5.2 POLYNOMIALS DEPENDENT ON ONE VARIABLE When a0, a1, a2, …, an are known numbers, the expression f (x) = a0 x n + a1 x n −1 + a2 x n − + + an −1 x + an (5.9) is said to be a polynomial, every xn–i, i = 0, 1, …, n summand at the right side of (5.9) is called the term of the polynomial Degree of highest order term of the polynomial is said to be degree of this polynomial It is clear that for a0 ≠ 0, the degree of f (x) equals n In (5.9), instead of x we write a certain number c, the obtained number f (c) is said to be the value of the polynomial at the point c When the value of the polynomial at the point c equals zero, the number c is said to be the root of this polynomial For example, as for the polynomial f(x) = x − x + x − x − , f (1) = 0, number is the root of this polynomial For any two polynomials f (x) and g (x) one can always find polynomials q (x) and r (x) with degree less than the degree of g (x) such that f (x) = g (x)q (x) + r (x) (5.10) Equation 5.10 expresses the theorem on division with remainder The following statement on division of polynomial into binomial, is true Theorem (Bezout) Necessary and sufficient condition for division of the polynomial f (x) into the binomial x – с is that the number с is the root of the polynomial f (x) The remainder obtained after dividing the polynomial f (x) into the binomial x – c equals f (c), that is, f (x) = ( x − c)q ( x) + f (c) (5.11) The following theorem resolves the problem on the existence of a root of each polynomial Theorem (the main theorem of algebra) Every polynomial with degree not less than unit has at least one root Higher Algebra Elements 251 Based on the main theorem of algebra and Bezout theorem, we can show that every polynomial with degree n (n ≥ 1) has rightly n number roots If the numbers с1, с2, …, сn are the roots of an n-th degree polynomial with higher term coefficient a0, then the equality f(x) = a0(x – c1)(x – c2) ⋅…⋅(x – cn) (5.12) is true If for the polynomial f(x) and the number c the equality f (x) = (x – c)k φ (x), φ (c) ≠ (5.13) is satisfied, c is called k times repeated root of f (x) Here k is a natural number satisfying the condition k ≤ n, φ (x) is the polynomial with degree not exceeding the degree of f (x) Once repeated, root is called a simple root When the complex number α = c + di is the root of real coefficient polynomial f (x), the complex number α = c−di adjoint to α is the root of f (x) It is known from the theorem on division with remainder that when dividing the n degree polynomial f (x) into the binomial x – c, the degree of quotient polynomial q (x) will be n − If we denote the coefficients of q (x) by b0, b1,…, bn−1, the remainder by r, we can find them by the following table called the Horner’s scheme: a0 a1 a2 … an−1 an b0 = a0 b1 = cb0+a1 b2 = cb1+a2 … bn−1 = cbn−2 + an−1 r = cb n−1 + a n The rule for using the table: we take b0 = a0, for finding the next element of every subsequent column of the second row, we multiply c by the antecedent column element of the second row and add to the same column element of the first row The Horner scheme is used for finding the value of a polynomial at certain point and to determine how many times the root of a polynomial is repeated Problems to be solved in auditorium Problem 450 Divide the polynomial f (x) into the polynomial g (x) with remainder: 252 Higher Mathematics for Engineering and Technology 1) f (x) = 3x5 + 7x4+12x3 + 17x2 + 7x + 2, g (x) = x3 + 2x2 + 3x + 4; 2) f (x) = x5 + 3x4 − 2x3 − 2x2 + 5x + 1, g (x) = x2 + 3x – Solution of 2): x5 + 3x4 − 2x3 − 2x2 + 5x + x2 + 3x − x5 + 3x4 − x3 x3 − x + −x3 − 2x2 + 5x + −x3 − 3x2 + x x2 + 4x + x2 + 3x − x+2 f (x) = g (x) (x3 − x +1) + (x + 2) Answer: 1) f (x) = g (x) (3x2+x+1) −2 Problem 451 Using the Horner scheme, divide the polynomial f (x) into the binomial g (x) with remainder: 1) f (x) = x + x − x + x − 5, g ( x) = x + 2; 2) f (x) = x + x − x + x − 3, g ( x) = x − Solution of 1): As x + = x – ( –2) in the Horner scheme we take c = –2, a0 = and get: −8 −5 (−2)⋅2+7=3 (−2)⋅3+0=−6 (−2)(−6)−8=4 (−2) ⋅4+3=−5 (−2)⋅(−5)−5=5 So, q(x) = x + 3x3 − x + x − 5, r = Therefore x5 + x − x + 3x − = ( x + 2)(2 x + x3 − x + x − 5) + Answer: 2) x + x − x + x − = ( x − 2)( x + x + x + 24) + 45 Problem 452 Determine how many times the root of the polynomial x0 of the number x0 is repeated: Higher Algebra Elements 253 1) f (x) = x − x + 10 x − x + 9, x0 = 3; 2) f (x) = x + x + 11x + x − 12 x − 8, x0 = −2 Solution of 1): By the Horner scheme, at first we divide f (x) into (x – x0) – a and then divide the obtained quotient polynomial again into (x – x0) – a and continue the process until we get a nonzero remainder The root x0 is repeated as many as the number of zeros 3 –6 10 –6 –3 –3 1 10 As zero remainders are obtained, x0 = is a twice-repeated root Answer: 2) three times repeated root Problem 453 Construct the least degree real coefficient polynomial with the given roots: 1) 2, 3, 1+i is a simple root, is a twice-repeated root; 2) 2−3i is a three times repeated root; 3) i is twice repeated, −1 is a simple root Answer: 1) c (x − 1) ( x − 2)( x − 3)( x − x + 2) = = c( x − x + 33 x − 65 x + 74 x − 46 x + 12); 2) c(x − x + 13)3 ; 3) c(x + 1) ( x + x − 2) = = c ( x + x + x + x + x + x + 2), c ≠ 0 are arbitrary real numbers Problem 454 Knowing that the number i is the root of the polynomial f(x) = x − x3 + x − x + , decompose this polynomial into linear cofactors Answer: (x + 1)( x − 2)( x − 3) 254 Higher Mathematics for Engineering and Technology Guideline: Taking into account that the number i is a root, divide the given polynomial into the polynomial x2 + Problem 455 Knowing that the number 1+i is the root of the equation x − x − x + x − = , find the other roots of this equation Answer: 1−i, −2, Home tasks Problem 456 Divide the polynomial f (x) into the polynomial g (x) with remainder: 1) f (x) = x − x − 12 x + 21x − 22 x + 7, g(x) = x + x − x + 4; 2) f (x) = x − x + x − x − 2, g(x) = x − x + x + Answer: 1) f (x) = g (x)(x − x + 1) + ( x + 3); 2) f (x) = g (x)(x − x − 1) + ( x + 1) Problem 457 Using the Horner scheme, divide the polynomial f (x) into the binomial g (x): 1) f (x) = x + x + x − 3, g (x) = x + 2; 2) f (x) = x − x − 38 x − x + 1, g (x) = x − Answer: 1) f (x) = g (x)(2 x − x + x) − 3; 2) f (x) = g (x)(5 x + 13 x + x + x + 5) + 16 Problem 458 Determine how many times the root of the polynomial f (x) of the number x0 is repeated: 1) f (x) = x − x + x − x + x − 8, x0 = 2; ) f (x) = x + x + 16 x3 + x − 16 x − 16 , x0 = −2 Answer: 1) is a three times repeated root; 2) is a four times repeated root Problem 459 Construct the least degree real coefficient polynomial with the given roots: 1) 2+i is a simple root, is a twice-repeated root: 2) 2−i is a twicerepeated root, −3 is a simple root Higher Algebra Elements 255 Answer: 1) c(x − x + 5)( x − 1) = c( x − x + x + x − 5); 2) c(x − x + 5) ( x + 3), c ≠ is an arbitrary real number Problem 460 Knowing that the number x0 is the root of the polynomial f (x), find the remaining roots of f (x): 1) f (x) = x − x + x − x + x − 1, x0 = is a three times repeated root; 2) f (x) = x + x + x + x + x + x + 2, x0 = i is a twice-repeated root Answer: 1) i and −i; 2) −i is a two times repeated root, −1 and −2 are simple roots KEYWORDS •• •• •• •• •• •• •• imaginary unit complex number adjoint complex number modulus argument simple root repeated root INDEX A Analytic geometry, elements, determinants and calculation, auxiliary diagonal, even substitution, feature, 9–10 n number, odd substitution, problems, 11–14 second-order, theorem, 10 inverse matrix and methods, 20 definition, 18–19 problems, 20–23 theorem, 19 transformations, 19 matrices and operations elements, intersection, law of permutation, multiplication, problems, 4–5 rectangle, mixed product of vectors definition, 54–55 formula, 55 problems, 55–58 operations on vectors in plane and space, 36 definition, 37–39 problems, 40–45 theorem, 37–39 plane and straight line equations in space, 78 coordinates, 75 general equation, 70, 71 normal vector, 71–72 normal vector of plane, 77 parallelism, 76 parametric equations, 70–71, 74 perpendicular, 77 piecewise equation, 72–73 problems, 78–95 vectorial equation, 74 rank of matrices and calculation rules definition, 14 k-th order determinant, 14 problems, 16–18 theorem, 15–16 scalar product of vectors definition, 45 problems, 46–49 second order curves asymptotes of hyperbola, 99 canonic equation, 96 definition, 95, 97–98, 100–101 directrix, 100 eccentricity, 98 ellipse, 97–98 equation of parabola, 101–102 hyperbola, 98–100 problems, 101–114 radius circle, 96 straight line tangent, 96–97, 97 straight line equations on plane, 59, 62 angular coefficient and equals, 59 canonical equation, 60 canonical equation of straight line, 60 equation of, 61 general equation of, 59 parametric equations, 60 piecewise equation, 62 problems, 62–69 258 Higher Mathematics for Engineering and Technology transform general equation to angular coefficient, 61 system of linear equations, 23 augmented matrix, 24 case, 26–28 compatible or joint system, 24 Gauss method, 25 generalized solution, 30 Kramer formula, 32 particular solutions, 29 problems, 31–35 theorem, 24–25 theorem (Kramer), 30–31 theorem (of Kronecker–Capelli), 24 vectorial product of vectors definition, 49–51 Auxiliary diagonal, B Bernoulli-de L’ Hospital rule, 211 problems, 213–219 validity of the equality, 212 C Canonical equation of straight line, 60 Complex numbers and operations division, 246 Euler formula, 246 imaginary unit, 244 modulus of, 245 multiplication, 246 notation, 244 problems, 247–250 raising to power, 246 rectangular coordinate system, 245 Convexity of graph of function direction and turning points, 230 definitions, 231–232 problems, 232–236 theorems, 231–232 D Derivative calculating rules definition, 184–185 first-order derivative, 187 function, 185 logarithmic derivative, 188 natural logarithm, 188 problems, 189–199 theorem, 185–186 differential, applications definition, 200 problems, 201–205 straight line perpendicular, 199 Determinants and calculation, auxiliary diagonal, even substitution, feature, 9–10 n number, odd substitution, problems, 11–14 second-order, theorem, 10 Direction and turning points, 230 definitions, 231–232 problems, 232–236 theorems, 231–232 E Euler formula, 246 Even substitution, F Function of one variable Bernoulli-de L’ Hospital rule, 211 problems, 213–219 validity of the equality, 212 derivative and differential, applications definition, 200 first-order derivative, 187 function, 185 logarithmic derivative, 188 natural logarithm, 188 problems, 189–199, 201–205 straight line perpendicular, 199 theorem, 185–186 259 Index theorems of differential calculus Maclaurin formula, 207 problems, 207–211 remainder term, 207 Taylor’s formula, 206 G General equation of straight line, 59 Graphing of function, 236 problems, 237–241 H Higher algebra elements complex numbers and operations division, 246 Euler formula, 246 imaginary unit, 244 modulus of, 245 multiplication, 246 notation, 244 problems, 247–250 raising to power, 246 rectangular coordinate system, 245 polynomials dependent on one variable degree of highest order, 250 problems, 251–255 theorem, 250–251 I Inverse matrix and methods, 20 definition, 18–19 problems, 20–23 theorem, 19 transformations, 19 K Kramer formula, 32 L Largest and least values of function extremum definition, 222–223 problems, 224–230 theorem, 222–223 monotonicity intervals definition, 222–223 problems, 224–230 theorem, 222–223 Linear algebra, elements determinants and calculation, auxiliary diagonal, even substitution, feature, 9–10 n number, odd substitution, problems, 11–14 second-order, theorem, 10 inverse matrix and methods, 20 definition, 18–19 problems, 20–23 theorem, 19 transformations, 19 matrices and operations elements, intersection, law of permutation, multiplication, problems, 4–6 rectangle, mixed product of vectors definition, 54 formula, 55 problems, 55–58 operations on vectors in plane and space, 36 definition, 37–39 problems, 40–45 theorem, 37–39 plane and straight line equations in space, 78 coordinates, 75 general equation, 70 parallelism, 76 parametric equations, 70–71 perpendicular, 77 260 Higher Mathematics for Engineering and Technology piecewise equation, 72–73 problems, 78–95 vectorial equation, 74 rank of matrices and calculation rules definition, 14 k-th order determinant, 14 problems, 188–199 theorem, 15–16 scalar product of vectors definition, 45 problems, 46–49 second order curves asymptotes of hyperbola, 99 canonic equation, 96 definition, 95, 97–98, 100–101 directrix, 100 eccentricity, 98 ellipse, 97–98 equation of parabola, 101–102 hyperbola, 98–100 problems, 101–114 radius circle, 96 straight line tangent, 96–97, 97 straight line equations on plane, 59, 62 angular coefficient and equals, 59 canonical equation, 60 equation of, 61 general equation of, 59 parametric equations, 60 piecewise equation, 62 problems, 62–69 transform general equation to angular coefficient, 61 system of linear equations, 23 augmented matrix, 24 case, 26–28 compatible or joint system, 24 Gauss method, 25 generalized solution, 30 Kramer formula, 32 particular solutions, 29 problems, 31–35 theorem, 24–25 theorem (Kramer), 30–31 theorem (of Kronecker–Capelli), 24 vectorial product of vectors definition, 49–51 M Matrices and operations elements, intersection, law of permutation, multiplication, problems, 4–5, 4–6 rectangle, Mixed product of vectors definition, 54–55 formula, 55 problems, 55–58 O Odd substitution, Operations on vectors in plane and space, 36 definition, 37–39 problems, 40–45 theorem, 37–39 P Parametric equations plane, 70 straight line, 60 Plane and straight line equations in space, 78 coordinates, 75 general equation, 70, 71 normal vector, 71–72, 77 parallelism, 76 parametric equations, 70–71, 74 perpendicular, 77 piecewise equation, 72–73 problems, 78–95 vectorial equation, 74 Polynomials dependent on one variable degree of highest order, 250 problems, 251–255 261 Index theorem, 250–251 R Rank of matrices and calculation rules definition, 14 k-th order determinant, 14 problems, 16–18, 188–199 theorem, 15–16 S Scalar product of vectors definition, 45 problems, 46–49 Second order curves asymptotes of hyperbola, 99 canonic equation, 96 definition, 95, 97–98, 100–101 directrix, 100 eccentricity, 98 ellipse, 97–98 equation of parabola, 101–102 hyperbola, 98–100 problems, 101–114 radius circle, 96 straight line tangent, 96–97, 97 Straight line equations on plane, 59, 62 angular coefficient and equals, 59 canonical equation, 60 equation of, 61 general equation of, 59 parametric equations, 60 piecewise equation, 62 problems, 62–69 transform general equation to angular coefficient, 61 System of linear equations, 23 augmented matrix, 24 case, 26–28 compatible or joint system, 24 Gauss method, 25 generalized solution, 30 Kramer formula, 32 particular solutions, 29 problems, 31–35 theorem, 24–25 theorem (Kramer), 30–31 theorem (of Kronecker–Capelli), 24 T Theorems of differential calculus Maclaurin formula, 207 problems, 207–211 remainder term, 207 Taylor’s formula, 206 V Vectorial product of vectors definition, 49–51 .. .HIGHER MATHEMATICS FOR ENGINEERING AND TECHNOLOGY Problems and Solutions www.Technicalbookspdf.com www.Technicalbookspdf.com HIGHER MATHEMATICS FOR ENGINEERING AND TECHNOLOGY Problems and Solutions. .. www.Technicalbookspdf.com 16 Higher Mathematics for Engineering and Technology the column number of nonzero two elements in the preceding row For example, the matrix is in the step form By elementary transformations... second addends in www.Technicalbookspdf.com 10 Higher Mathematics for Engineering and Technology the same row and column of the second determinant, the remaining rows and columns of both determinants