Lewis Carroll’s Condensation of Determinant Applied in Solving Linear Systems and Inverting Matrices

Contents

Preface

1 Introduction
1.1 Linear Systems
1.2 Matrices
1.3 Determinant Notation
1.4 Inverses of Matrices

2 Dodgson’s Condensation
2.1 Dodgson’s Life
2.2 Condensation Method
2.3 Present Interest in Dodgson’s condensation

3 DivisionbyZero Problem
3.1 Recommencing the Operation Method
3.2 Limit Method

4 Solving Linear Systems
4.1 Cramer’s Rule
4.2 A New Method using Dodgson’s condensation

5 InversesofMatrices
5.1 Matrix Inversion by Minors
5.2 Matrix Inversion by Gauss–Jordan method
5.3 A New Method of Matrix Inversion
5.4 Proof of the Validity of the New Approach
5.4.1 Derivation for 2 × 2 Matrix
5.4.2 Derivation for 3 × 3 Matrix
5.4.3 Derivation for 4 × 4 Matrix

Bibliography

Preface

He author’s chief purpose of writing this little book is to expose the reader to new applications of Dodgson’s condensation for deter­minant evaluation, namely the solving of large linear systems and the computation of the inverses of large matrices. In the course of discussing these applications, the Author attempts to make the work of “the great Ox­ford lecturer”, Rev. Charles Lutwidge Dodgson (1832–1898), accessible to the mathematician of today who might not be able either to read his 1866 paper delivered to the Royal Society of London, or, having read it, to master it and grasp the whole scheme of the paper. I feel that I owe even less of an apology for offering to the public an extract of the extant work of one of the greatest mathematical geniuses that the world have ever had.

Dodgson’s condensation is a well-known technique in linear algebra and is open to investigations. Although some mathematicians have recently given much attention to its investigations and their results cover a significant area of linear algebra, this method is still so far from being exhausted, that it can well be said that, up to this time, but an extremely small portion of an exceedingly fruitful aspect of Dodgson’s condensation has been cultivated.

To find possible applications of Dodgson’s condensation, the author sought few years ago to give a new phase to this study. An object of the present discussion is further to open up other new results and to develop some of the new truths which thus become accessible. We shall here give an account of those things which can be made intelligible in a few words. Thus the author has embodied in the present book a part of his investigations of Dodgson’s condensation, and this part may, to some extent, be generally helpful in many further investigations.

The arrangement of this book is as follows: The first chapter introduces linear systems in relation to matrices, determinants and inverses of matrices. The second chapter introduces Dodgson’s condensation as a method of eval­uating the determinants of matrices. The third chapter is concerned with the methods of overcoming the problem of division by zero which is sometimes encountered when finding the determinants of matrices using Dodgson’s con­densation. The fourth chapter treats the application of the condensation in solving linear systems of equations while the fifth and last chapter is intended to give the reader another application of the condensation, the computation of the inverses of large matrices. The treatment throughout is made as brief as is possible consistent with clearness and is confined entirely to fundamen­tal matters. This is done because it is believed that in this manner the book may best be made to serve its purpose in exposing the reader to the uses of Dodgson’s Condensation in Linear Algebra.

I proceed to make a single remark on the source from which the portions of this book is made up. I should add that this book is made up from a research work on Dodgson’s condensation made by me a few years ago; and the new approaches introduced are my own discoveries resulting from the research.

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1.1 Linear Systems

A linear system, which often arises in physical science, engineering, and busi­ness, is a set or system of two or more linear equations containing common variables or unknowns. A general linear system of n simultaneous equations with n unknown variables can be written as

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Lewis Carroll ’ s Condensation of Determinant the system, and b 1, b 2, b 3, . . . , bn are the constant terms. A solution of the system is a set of values of the unknowns that satisfies every equation of the system simultaneously.

1.2 Matrices

An efficient and compact way of solving a linear system is by expressing it as a matrix, a rectangular array of numbers arranged in rows and columns and enclosed in brackets. The system (4.6) of linear equations can be written compactly in matrix form as Anx = b where

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By augmenting An by column b. The matrix containing the coefficients and constants of the system (4.6) is called the augmented matrix of the system and the equation (1.5) is called the augmented matrix equation of the system. The augmented matrix equation (1.5) determines the system (4.6) completely since it contains all the numbers in the system. The rows of any augmented matrix equation can be treated just like the equations of the corresponding system of linear equations. So, performing elementary row operations on the augmented matrix of a system does not alter its solution.

1.3 Determinant Notation

The notation of determinant arises whenever the unknowns of a system of linear equations are eliminated.This term, though impractical in computa­tions , has valuable applications in eigenvalue problems, differential equa­tions, vector algebra, optimization, and so on. Now a determinant is an array of numbers written as a square, and denoted by two vertical lines enclosing the array. The determinant of the nxn matrix An (4.7), called an n th order determinant, is denoted as |An| and written as

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The third-order determinant is, thus, the summation of three products, each product being the multiplication of a single number in the first row and its minor, a second-order determinant. In like manner, the fourth-order deter­minant is the summation of four products, each product being the multipli­cation of a single number in the considered row and its minor, a third-order determinant, which in turn is broken down into second-order determinants. Higher order determinants are treated in a similar fashion.

Although 2nd order determinant can be evaluated very easily, expanding higher order determinants using the normal expansion along a row or column is lengthy, painful, tedious, cumbersome, error-prone and time-consuming. The expansion of third-order determinants using the normal expansion along a row or column is a little bit bearable. As the order of a determinant grows

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1.4 Inverses of Matrices

The scientist has long been plagued with large linear systems of equations. This problem lends itself to matrix analysis and the approach usually em­ployed is matrix inversion. If we start with the already mentioned n th order matrix equation lthough the real purpose of this work is the discussion of new applications of Dodgson’s condensation, namely, solving large lin­ear systems and computing inverses of large matrices, nevertheless we shall first develop what is already known, partly for the convenience of the reader, and partly because our method of treatment is different from that which has hitherto been employed.

2 Dodgson’s Condensation

2.1 Dodgson’s Life

Dodgson’s condensation, to which this book is wholly devoted, is one of the most enchanting mathematical discoveries of the 19th century. But, before discussing it, we shall give a brief sketch of Dodgson’s life.

One remarkable feature of the reign of the illustrious Queen Victoria of England (1819 - 1901), who is associated with Britain’s great age of indus­trial expansion, economic progress and, especially, empire, was the appear­ance of a genius, Britain’s famous Lewis Carroll. Charles Lutwidge Dodgson, the son of a clergyman, the third of 11 children, all of whom stuttered, is truly Lewis Carroll writing under a pseudonym (pen name). He was born in England in 1832 and when he was a child, he was fascinated with mathe­matics and solving puzzles. As a young man he was not comfortable in the company of adults but greatly enamoured with young girls and is said to have spoken without stuttering only to them, many of whom he entertained, corresponded with, and photographed. Although he was close to young girls, he was extremely puritanical and religious.

Dodgson graduated from Oxford in 1854, was appointed lecturer in math­ematics at Christ Church College, Oxford, in 1855, obtained his master of arts degree in 1857, and was ordained in the Church of England in 1861. In later years, he spent most of his time in writing, solving puzzles, and teach­ing at Christ Church. His writings include poetry, puns, novels, articles, and books on geometry, determinants, and the mathematics of tournaments and elections.

His intimacy with Dean Henry Liddell’s three young daughters, Lorina, Edith, and Alice Liddell, contributed to his writing of his excellent nursery tale Alice’s Adventures in Wonderland (1865) which made him famous and brought him a lot of money. Queen Victoria told Dodgson of her relish in reading Alice’s Adventures in Wonderland and how much she wanted to read his next book; he is said to have sent her Symbolic Logic, one of his most celebrated mathematical works. Late in life, Dodgson denied that he and Carroll were the selfsame person, even though he distributed hundreds of signed copies of the Alice’s Adventures in Wonderland to children and children’s hospitals.

In early January 1898, Dodgson was struck with a cold which at first seemed very minor but developed into a chest problem, which terminated the illustrious career of this wonderful genius, at the age of sixty-five.

2.2 Condensation Method

It was in order to hand-evaluate with ease and without pains the determinant of higher order that an interesting and attractive technique was introduced by Dodgson, namely the condensation method. The condensation method is a method of hand-solving simultaneous linear equations with large number of unknowns and hand-computing the determinants of large orders. The ba­sic idea of the condensation method is to reduce or condense an n th order matrix equation or determinant to an (n- 1)st order matrix equation or de terminant. Repeated application of the method results in a 2nd order matrix equation which can be easily solved or a 2nd order determinant which can be easily evaluated. Usually the (n - 1)st order matrix equation or determinant consists of numbers obtained by evaluating 2nd order determinants created by the entries of the n th order matrix equation or determinant.

Dodgson made many significant mathematical discoveries. Of these, it is his elegant condensation method that is arguably the most notable, a technique for which he deserves to be esteemed in the world of mathematics, especially in linear algebra.

Now Dodgson’s condensation consists of the following steps or rules:

1. Employ the elementary row and column operations to rearrange, if necessary, the given n th order matrix such that there are no zeros in its interior. The interior of a matrix is the minor formed after the first and last rows and columns of the matrix have been deleted.
2. Evaluate every 2nd order determinant formed by four adjacent el­ements. The values of the determinants form the (n - 1)st order matrix.
3. Condense the (n - 1)st order matrix in the same manner, dividing each entry by the corresponding element in the interior of the n th order matrix.
4. Repeat the condensation process until a single number is obtained. This number is the value of the determinant of the n th order matrix.

To make the method clear, we consider the matrix

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This in turn, by rule 3, is condensed to give the value, 7. Dividing this value by the interior, 1, of the 3rd order matrix, we get 7 which is the value of the determinant of our original 3rd order matrix.

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