284,99 €
Mehr erfahren.
- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Serie: Enzyme Reaction Engineering
- Sprache: Englisch
Mathematics for Enzyme Reaction Kinetics and Reactor Performance is the first set in a unique 11 volume-collection on Enzyme Reactor Engineering. This two volume-set relates specifically to the wide mathematical background required for systematic and rational simulation of both reaction kinetics and reactor performance; and to fully understand and capitalize on the modelling concepts developed. It accordingly reviews basic and useful concepts of Algebra (first volume), and Calculus and Statistics (second volume). A brief overview of such native algebraic entities as scalars, vectors, matrices and determinants constitutes the starting point of the first volume; the major features of germane functions are then addressed. Vector operations ensue, followed by calculation of determinants. Finally, exact methods for solution of selected algebraic equations - including sets of linear equations, are considered, as well as numerical methods for utilization at large. The second volume begins with an introduction to basic concepts in calculus, i.e. limits, derivatives, integrals and differential equations; limits, along with continuity, are further expanded afterwards, covering uni- and multivariate cases, as well as classical theorems. After recovering the concept of differential and applying it to generate (regular and partial) derivatives, the most important rules of differentiation of functions, in explicit, implicit and parametric form, are retrieved - together with the nuclear theorems supporting simpler manipulation thereof. The book then tackles strategies to optimize uni- and multivariate functions, before addressing integrals in both indefinite and definite forms. Next, the book touches on the methods of solution of differential equations for practical applications, followed by analytical geometry and vector calculus. Brief coverage of statistics-including continuous probability functions, statistical descriptors and statistical hypothesis testing, brings the second volume to a close.
Sie lesen das E-Book in den Legimi-Apps auf:
Seitenzahl: 1254
Veröffentlichungsjahr: 2020
Ähnliche
Enzyme Reactor Engineering: Forthcoming Titles
ANALYSIS OF ENZYME REACTOR PERFORMANCE
Volume 1
1. Ideal Reactors: Single Unit
Volume 2
2. Ideal Reactors: Multiple Units
Volume 3
3. Nonideal Reactors: Homogeneous with Convection
4. Nonideal Reactors: Homogeneous and Heterogeneous with Diffusion
Volume 4
5. Integration of Chemical Reaction and Physical Separation
Volume 5
6. Integration of Chemical Reactor and External Control
ANALYSIS OF ENZYME REACTION KINETICS
Volume 1
1. Mathematical Approach to Rate Expressions
2. Statistical Approach to Rate Expressions
Volume 2
3. Physical Modulation of Reaction Rate
4. Chemical Modulation of Reaction Rate
ENZYME REACTION KINETICS AND REACTOR PERFORMANCE
Volume 1
1. Basic Concepts of Reactions and Reactors
2. Basic Concepts of Hydrodynamics
Volume 2
3. Basic Concepts of Mass Transfer
4. Basic Concepts of Enthalpy Transfer
5. Basic Concepts of Chemical Reaction
6. Basic Concepts of Enzymes
Mathematics for Enzyme Reaction Kinetics and Reactor Performance
Volume 1
F. Xavier Malcata
Department of Chemical Engineering, University of PortoPortugal
You cannot teach a man anything; you can only help him find it for himself.
Galileo Galilei
This edition first published 2019© 2019 John Wiley & Sons Ltd
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions.
The right of F. Xavier Malcata to be identified as the author of this work has been asserted in accordance with the law.
Registered OfficesJohn Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USAJohn Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK
Editorial OfficeThe Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK
For details of our global editorial offices, customer services, and more information about Wiley products visit us at www.wiley.com.
Wiley also publishes its books in a variety of electronic formats and by print‐on‐demand. Some content that appears in standard print versions of this book may not be available in other formats.
Limit of Liability/Disclaimer of WarrantyIn view of ongoing research, equipment modifications, changes in governmental regulations, and the constant flow of information relating to the use of experimental reagents, equipment, and devices, the reader is urged to review and evaluate the information provided in the package insert or instructions for each chemical, piece of equipment, reagent, or device for, among other things, any changes in the instructions or indication of usage and for added warnings and precautions. While the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.
Library of Congress Cataloging‐in‐Publication Data
Names: Malcata, F. Xavier, author.Title: Mathematics for enzyme reaction kinetics and reactor performance / F. Xavier Malcata, Department of Chemical Engineering, University of Porto, Portugal.Description: Hoboken : Wiley, [2019‐] | Series: Enzyme reaction engineering | Includes bibliographical references and index. |Identifiers: LCCN 2018022263 (print) | LCCN 2018028979 (ebook) | ISBN 9781119490326 (Adobe PDF) | ISBN 9781119490333 (ePub) | ISBN 9781119490289 (volume 1 : hardcover)
Subjects: LCSH: Enzyme kinetics–Mathematics.Classification: LCC QP601.3 (ebook) | LCC QP601.3 .M35 2019 (print) | DDC 572/.744–dc23LC record available at https://lccn.loc.gov/2018022263
Cover Design: Wiley
Cover Image: Background © Zarya Maxim Alexandrovich/Shutterstock, Foreground (left) © Laguna Design/Getty Images, (right) Courtesy of F. Xavier Malcata
To my family: António, Mª Engrácia, Ângela, Filipa, and Diogo.For their everlasting understanding, unselfish support, and endless love.
About the Author
Par est scientia laboris.
(Work is ever the mate of science.)
Prof. F. Xavier Malcata was born in Malange (Angola) in 1963, and earned: a B.Sc. degree in Chemical Engineering (5‐year program), from the University of Porto (UP, Portugal) in 1986 (with first class honors); a Ph.D. degree in Chemical Engineering (with a distributed minor in Food Science, Statistics and Biochemistry), from the University of Wisconsin (UW, USA) in 1991; an equivalent Doctoral degree in Biotechnology – food science and technology, from the Portuguese Catholic University (UCP, Portugal) in 1998; and a Habilitation degree in Food Science and Engineering, also from UCP, in 2004.
Prof. Malcata has held academic appointments as: Teaching Assistant at UCP in 1985–1987 and at UW in 1988; Lecturer at UW in 1989; Assistant Professor at UCP in 1991–1998; Associate Professor at UCP in 1998–2004; and Full Professor at UCP in 2004–2010, Superior Institute of Maia (ISMAI, Portugal) in 2010–2012, and UP since 2012. He also held professional appointments as: Dean of the College of Biotechnology of UCP in 1998–2008; President of the Portuguese Society of Biotechnology in 2003–2008; Coordinator of the Northern Chapter of Chemical Engineering of the Portuguese Engineering Accreditation Board in 2004–2009; Official Delegate, in 2002–2013, of the Portuguese Government to the VI and VII Framework Programs of R&D held by the European Union – in such key areas as food quality and safety, and food, agriculture (including fisheries), and biotechnology, respectively; Chief Executive Officer of the University/Industry Extension (nonprofit) Associations AESBUC in 1998–2008 and INTERVIR+ in 2006–2008; and Chief Executive Officer of the Entrepreneurial Biotechnological Support Associations CiDEB in 2005–2008 and INOVAR&CRESCER in 2006–2008.
Over the years, the author has received several national and international public recognitions and awards, including: Cristiano P. Spratley Award by UP, in 1985; Centennial Award by UP, in 1986; election for membership in Phi Tau Sigma – honor society of food science (USA), in 1990; election for Sigma Xi – honor society of scientific and engineering research (USA), in 1990; election for Tau Beta Pi – honor society of engineering (USA), in 1991; Ralph H. Potts Memorial Award by American Oil Chemists’ Society (AOCS, USA), in 1991; election for New York Academy of Sciences (USA), in 1992; Foundation Scholar Award – dairy foods division by American Dairy Science Associaton (ADSA, USA), in 1998; decoration as Chevalier dans l’Ordre des Palmes Académiques by French Government, in 1999; Young Scientist Research Award by AOCS, in 2001; Canadian/International Constituency Investigator Award in Physical Sciences and Engineering by Sigma Xi, in 2002 and 2004; Excellence Promotion Award by Portuguese Foundation for Science and Technology (Portugal), in 2005; Danisco International Dairy Science Award by ADSA, in 2007; Edgar Cardoso Innovation Award by the Mayor of Gaia, in 2007; Scientist of the Year Award by European Federation of Food Science and Technology (Netherlands), in 2007; Samuel C. Prescott Award by Institute of Food Technologists (IFT, USA), in 2008; International Leadership Award by International Association of Food Protection (IAFP, USA), in 2008; election for Fellow by IFT, in 2011; Elmer Marth Educator Award by IAFP, in 2011; election for Fellow by International Academy of Food Science and Technology (IAFoST), in 2012; Distinguished Service Award by ADSA, in 2012; election for Fellow by ADSA, in 2013; J. Dairy Sci. Most Cited Paper Award by ADSA, in 2012; William V. Cruess Award for excellence in teaching by IFT, in 2014; and election for Fellow by AOCS, in 2014.
Among his many scientific interests, Prof. Malcata has focused his research chiefly on four major areas: theoretical simulation and optimization of enzyme reactors, theoretical optimization of thermodynamically and kinetically controlled processes, production and immobilization of oxidoreductases and hydrolases for industrial applications, and design and optimization of bioreactors to produce and process edible oils. In addition, he has developed work on: microbiological and biochemical characterization and technological improvement of traditional foods, development of nutraceutical ingredients and functional foods, rational application of unit operations to specific agri‐food processing, and design and development of novel photobioreactors for cultivation of microalgae, aimed at biofuel or high added‐value compound production. To date, he has published more than 400 papers in peer‐reviewed international journals that received more than 12000 official citations in all (without self-citations), corresponding to an h‐index of 54; he has supervised 30 Ph.D. dissertations successfully concluded; he has written 14 monographs and edited 5 multiauthored books; he has authored more than 50 chapters in edited books and 35 papers in trade journals, besides more than 50 technical publications. He was also a member of about 60 peer‐reviewing committees of research projects and fellowships; he has acted as supervisor of 90 individual fellowships, most at Ph.D. and postdoctoral levels, and collaborated in 60 research and development projects – of which he has served as principal investigator in 36; he has participated in 50 organizing/scientific committees of professional meetings; he has delivered 150+ invited lectures worldwide, besides almost 600 volunteer presentations in congresses and workshops; he has served in the editorial board of 5 major journals in the applied biotechnology, and food science and engineering areas; and he has reviewed several hundred manuscripts for journals and encyclopedia. He has been a longstanding member of American Institute of Chemical Engineers, American Chemical Society, IFT, American Association for the Advancement of Science, AOCS, IAFP, and ADSA.
Series Preface
Ad augusta per angusta.
(Toward the top, through hard work)
Comprehensive mathematical simulation – using mechanistic models as far as possible, constitutes an essential contribution to rationally characterize performance, as well as support design and drive optimization of any enzyme reactor. However, too often studies available in the literature – including text and reference books, deal with extensive modelling of chemical reactors that employ inorganic catalysts, or instead present extensive kinetic analysis of enzymes acting only (and implicitly) in batch apparatuses. Although constraining from an engineering perspective, this status quo is somewhat expected – because chemical engineers typically lack biochemical background, while biochemists miss engineering training. Meanwhile, rising environmental concerns and stricter legislation worldwide have urged the industry to resort to more sustainable, efficient, and cleaner processes – which tend to mimic natural (i.e. enzyme‐mediated) pathways; they generate essentially no polluting effluents or residues, require mild conditions of operation, and exhibit low‐energy requirements – while taking advantage of the extremely high activity and unique substrate selectivity of enzymes. The advent of genetic engineering has also dramatically contributed to drop the unit price, and enlarge the portfolio of enzymes available for industrial purposes, via overexpression in transformed microorganisms and development of sophisticated purification techniques; and advances in molecular engineering have further permitted specific features, in terms of performance and stability, be imparted to enzymes for tailored uses, besides overcoming their intrinsic susceptibility to decay.
An innovative approach is thus in order, where fundamental and applied aspects pertaining to enzyme reactors are comprehensively tackled – built upon mathematical simulation, and encompassing various ideal and nonideal configurations, presented and discussed in a consistent and pragmatic way. Enzyme Reactor Engineering pursues this goal, and accordingly conveys the most integrated and complete treatment of the subject of enzyme reactors to date; it will likely materialize a qualitative leap toward more effective strategies of describing, designing, and optimizing said reactors. More than a mere description of technology, true engineering aspects departing from first principles are put forward, and their rationale is systematically emphasized – with special attention paid to stepwise derivation of the underlying equations, so as to permit a self‐paced learning program by any student possessing elementary knowledge of algebra, calculus, and statistics. A careful selection of mathematical tools deemed useful for enzyme reactors is also provided in dedicated volumes, for the more inquisitive students and practitioners – in a straightforward, yet fully justified manner. Furthermore, appropriate examples, based (at least) on Michaelis and Menten’s enzymatic kinetics and first‐order enzyme decay, are worked out in full – for their being representative of industrial situations, while exhibiting a good compromise between practical applicability and mathematical simplicity. In this regard, the present book collection represents an unparalleled way of viewing enzyme reactors – clearly focused on the reactor component but prone to build an integrated picture, including mixture via momentum and mass transfer, and subsequent transformation via chemical reaction, with underlying enthalpic considerations as found necessary.
In a word, Enzyme Reactor Engineering attempts to contribute to a thorough understanding of the engineering concepts behind enzyme reactors – framed by a rigorous mathematical and physically consistent approach, and based on mechanistic expressions describing physical phenomena and typical expressions for enzyme‐mediated kinetics and enzyme decay. It takes advantage of a multiplicity of mathematical derivations, but ends up with several useful formulae while highlighting general solutions; and covers from basic definitions and biochemical concepts, through ideal models of flow, eventually to models of actual reactor behavior – including interaction with physical separation and external control. The typical layout of each chapter accordingly includes: introductory considerations, which set the framework for each theme in terms of relevance; objective definition, which entails specific goals and usefulness of ensuing results; and mathematical stepwise development, interwoven with clear physicochemical discussion (wherever appropriate), which resort to graphical interpretations and present step‐by‐step proofs to eventually generate (duly highlighted) milestone formulae. All in all, such an approach is aimed at helping one grasp the essence of descriptive functions, as well as the meaning behind hypothesized parameters and attained optima. Selected papers, chapters, or books are listed at the end, for more in‐depth, complementary reading – aimed at reinforcing global overviews.
Enzyme Reactor Engineering is organized as four major sets, which support a self‐consistent and ‐contained book collection: Enzyme Reaction Kinetics and Reactor Performance, Analysis of Enzyme Reaction Kinetics, Analysis of Enzyme Reactor Performance, and Mathematics for Enzyme Reaction Kinetics and Reactor Performance. Such a philosophy is primarily intended to help the prospective learner evolve in their knowledge acquisition steps – although it also constitutes standard material suitable for instructors; and allows the reader to first grasp the supporting concepts before proceeding to a deeper and deeper insight on the detailed kinetics of reactions brought about by generic enzymes, and eventually extending said concepts to overall reactor operation using enzymes. Three levels of description are indeed apparent and sequentially considered in the core of this book collection: macroscopic, or ideal; microscopic, or nonideal in terms of hydrodynamics (including homogeneous, nontrivial flow patterns) and mass transfer (including multiphasic systems); and submicroscopic, or nonideal in terms of mixing. The quality of the approximation increases in this order – but so does the complexity of the mathematical models entertained, and the thoroughness of the experimental data required thereby.
This treatise on reactors, using enzymes as catalysts, should be usable by and useful to both (advanced) undergraduate and graduate students interested in the fascinating field of white biotechnology, and typically enrolled in chemical or biochemical engineering degrees; as well as industrial practitioners involved in conceptual design or concerned with rational optimization of enzyme‐mediated processes. Enzyme Reactor Engineering has been conceived for hybrid utilization as both text and reference – since virtually all topics of relevance for enzyme reactors are addressed to some extent, and carefully related to each other.
F. Xavier MalcataProfessor of Chemical EngineeringUniversity of Porto (Portugal)
Preface
Quality is not an act, it is a habit.
Aristotle
Mathematics for Enzyme Reaction Kinetics and Reactor Performance is the first set in a unique 11‐volume collection on Enzyme Reactor Engineering. This two‐volume set relates specifically to the mathematical background – required for systematic and rational simulation of both reaction kinetics and reactor performance, and to fully understand and capitalize on the modelling concepts developed; it accordingly reviews basic and useful concepts of Algebra (first volume), and Calculus and Statistics (second volume).
A brief overview of such native algebraic entities as scalars, vectors, matrices, and determinants constitutes the starting point of the first volume; the major features of germane functions are then addressed – namely, polynomials and series (and their operative algebra), as well as trigonometric and hyperbolic functions. Vector operations ensue, with results either of scalar or vector nature, complemented by tensor/matrix operations and their properties. The calculation of determinants is considered next – with an emphasis on their underlying characteristics, and use to find eigenvalues and -vectors. Finally, exact methods for solution of selected algebraic equations, including sets of linear equations, are addressed – as well as numerical methods for utilization at large.
The second volume departs from introduction of seminal concepts in calculus, i.e. limits, derivatives, integrals, and differential equations; limits, along with continuity, are further expanded afterward, covering uni‐ and multi‐variate cases, as well as classical theorems. After recovering the concept of differential and applying it to generate (regular and partial) derivatives, the most important rules of differentiation of functions (in explicit, implicit, and parametric forms) are retrieved – and the most relevant theorems supporting simpler manipulation thereof are reviewed. Once the conditions for independence of functions are put forward, the strategies to optimize uni‐ and multi-variate functions are tackled – either in the presence or absence of constraints. The concept of integral is finally discussed, in both indefinite and definite forms – and the fundamental theorems are brought on board, along with the rules of integration. Furthermore, optimization of integrals is discussed, as part of calculus of variations. Practical applications of the concept of derivative follow – namely, for development of Taylor’s series and setting of associated convergence criteria; and also of the concept of integral – namely, to define the gamma function and to take advantage of its ubiquitous properties. Due to their relevance in reactor modelling, fundamental concepts in analytical geometry are recalled – with an emphasis on curves, surfaces, and volumes bearing simple (yet representative) shapes. Finally, the working horse of process modelling is covered to some length – i.e. (ordinary) differential equations, including such useful tools for solution thereof as Laplace’s integral (and Legendre’s derivative) transforms – with a brief excursion to solution of (first order) partial differential equations. The most important methods of analytical solution available are duly reviewed and eventually complemented with simple numerical approaches to integration. The final topic is vector calculus – where the nuclear del operator is introduced, and the most important applications to scalar and vector entities are developed as identities expressed in rectangular coordinates; an extension is made later to cylindrical and spherical coordinates, for the sake of completeness.
The second volume ends with a brief coverage of statistics – starting with continuous probability functions and statistical descriptors, and proceeding to discussion in depth of the normal distribution; such other continuous distributions as lognormal, chi‐square, Student's t‐, and Fisher's F‐distributions are reviewed next – spanning from mathematical derivation, through calculation of major descriptors, to discussion of most relevant features (including generation of distinct continuous probability functions). Statistical hypothesis testing is addressed next, complemented with the alternative approach of parameter and prediction inference – resorting to linear regression analysis as germane mode of parameter estimation.
F. Xavier MalcataProfessor of Chemical EngineeringUniversity of Porto (Portugal)
Volume 1
1Basic Concepts of Algebra
Reading maketh a full man, conference; a ready man, and writing; an exact man.
Francis Bacon
1Scalars, Vectors, Matrices, and Determinants
Quantification of any entity or concept requires association to a numerical scale, so as to permit subsequent abstract reasoning and objective comparability; hence, every measurement carried out in the physicochemical world leads to a number, or scalar. Such numbers may be integer, rational (if expressible in the form p/q, where p and q denote integer numbers), or irrational (if not expressible in the previous form, and normally appearing as an infinite, nonrecurring decimal). If considered together, rational and irrational numbers account for the whole of real numbers – each one represented by a point in a straight line domain.
Departing from real numbers, related (yet more general) concepts have been invented; this includes notably the complex numbers, z – defined as an ordered pair of two real numbers, say, z ≡ a + ιb, where a and b denote real numbers and ι denotes , the imaginary unit. Therefore, z is represented by a point in a plane domain. In the complex number system, a general nth degree polynomial equation holds exactly (and always) n roots, not necessarily distinct though – as originally realized by Italian mathematicians Niccolò F. Tartaglia and Gerolamo Cardano in the sixteenth century; many concepts relevant for engineering purposes, originally conceived to utilize real numbers (as the only ones adhering to physical evidence), may be easily generalized via complex numbers.
The next stage of informational content is vectors – each defined by a triplet (a,b,c), where c also denotes a real number; each one is represented by a point in a volume domain and is often denoted via a bold, lowercase letter (e.g. v). Their usual graphical representation is a straight, arrowed segment linking the origin of a Cartesian system of coordinates to said point – where length (equal to , as per Pythagoras’ theorem), coupled with orientations (as per tan{b/a} and tan{c/}) fully define the said triplet. An alternative representation is as [a b c] or – also termed row vector or column vector, respectively; when three column vectors are assembled together, say, , , and , a matrix results, viz. , termed tensor – which may also be obtained by joining three row vectors, say, [a1 a2 a3], [b1 b2 b3], and [c1 c2 c3]. The concept of matrix may be generalized so as to encompass other possibilities of combination of numbers besides a (3 × 3) layout; in fact, a rectangular (p × q) matrix of the form , or [ai,j ; i = 1, 2,…, p; j = 1, 2, …, q] for short, may easily be devised.
Matrices are particularly useful in that they permit algebraic operations (and the like) be performed once on a set of numbers simultaneously – thus dramatically contributing to bookkeeping, besides their help to structure mathematical reasoning. In specific situations, it is useful to design higher order number structures, such as arrays of (or block) matrices; for instance, may be also represented as , provided that, say, A1,1 ≡ [a1], A1,2 ≡ [a2 a3], , and represent, in turn, smaller matrices. An issue of compatibility arises in terms of the sizes of said blocks, though; for a starting (p × q) matrix A, only (p1 × q1) A1,1, (p1 × q2) A1,2, (p2 × q1) A2,1, and (p2 × q2) A2,2 matrices are allowed – obviously with p1 + p2 = p and q1 + q2 = q.
One of the most powerful applications of matrices is in solving sets of linear algebraic equations, say,
and
in its simplest version – where a1,1, a1,2, a2,1, a2,2, b1, and b2 denote real numbers, and x1 and x2 denote variables; if a1,1≠ 0 and a1,1a2,2− a1,2a2,1≠ 0, then one may start by isolating x1 in Eq. (1.1) as
and then replace it in Eq. (1.2) to obtain
After factoring x2 out, Eq. (1.4) becomes
so isolation of x2 eventually gives
– which yields a solution only when a1,1a2,2− a1,2a2,1≠ 0; insertion of Eq. (1.6) back in Eq. (1.3) yields
thus justifying why a solution for x1 requires a1,1≠ 0, besides a1,1a2,2− a1,2a2,1≠ 0 (as enforced from the very beginning). Equation (1.6) may be rewritten as
– provided that one defines
complemented with
the left‐hand sides of Eqs. (1.9) and (1.10) are termed (second‐order) determinants. If both sides of Eq. (1.2) were multiplied by −a1,2/a2,2, one would get
– so ordered addition of Eqs. (1.1) and (1.11) produces simply
after having x1 factored out; upon multiplication of both sides by a2,2, Eq.,(1.12) becomes
with isolation of x1 unfolding
– a result compatible with Eq. (1.7), once the two fractions are lumped, a1,2a2,1b1 canceled out with its negative afterward, and a1,1 finally dropped from numerator and denominator. Recalling Eq. (1.10), one may redo Eq. (1.14) to
as long as
is put forward; all forms conveyed by Eqs. (1.9), (1.10), and (1.16) do indeed share the form
irrespective of the values taken individually by α1,1, α1,2, α2,1, and α2,2. This is why the concept of determinant was devised – representing a scalar, bearing the unique property that its calculation resorts to subtraction of the product of elements in the secondary diagonal from the product of elements in the main diagonal of the accompanying (2 × 2) matrix. In the case of Eq. (1.10), the representation is selected because the underlying set of algebraic equations, see Eqs. (1.1) and (1.2), holds indeed as coefficient matrix. If a set of p algebraic linear equations in p unknowns is considered, viz.
then the concept of determinant can be extended in very much the same way to produce
