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You may copy it, give it away or % +% re-use it under the terms of the Project Gutenberg License included % +% with this eBook or online at www.gutenberg.org % +% % +% % +% Title: Condensation of Determinants, Being a New and Brief Method for Computing their Arithmetical Values +% % +% Author: Lewis Carroll (AKA Charles Lutwidge Dodgson) % +% % +% Release Date: September 8, 2011 [EBook #37354] % +% % +% Language: English % +% % +% Character set encoding: ISO-8859-1 % +% % +% *** START OF THIS PROJECT GUTENBERG EBOOK CONDENSATION OF DETERMINANTS *** +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\def\ebook{37354} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Packages and substitutions: %% +%% %% +%% book: Required. %% +%% inputenc: Standard DP encoding. Required. %% +%% %% +%% amsmath: AMS mathematics enhancements. Required. %% +%% amssymb: Additional mathematical symbols. Required. %% +%% %% +%% alltt: Fixed-width font environment. Required. %% +%% array: Enhanced tabular features. Required. %% +%% %% +%% geometry: Enhanced page layout package. Required. %% +%% fancyhdr: Enhanced page headers. Required. %% +%% hyperref: Hypertext embellishments for pdf output. Required. %% +%% ifthen: Logical conditionals. Required. %% +%% %% +%% %% +%% Producer's Comments: %% +%% %% +%% Changes are noted in this file with \DPtypo{}{}, showing %% +%% original and replacement text side-by-side. %% +%% %% +%% PDF pages: 20 %% +%% PDF page size: 5.25 x 8" (non-standard) %% +%% PDF document info: filled in %% +%% %% +%% %% +%% Command block: %% +%% %% +%% pdflatex x2 %% +%% %% +%% %% +%% %% +%% September 2011: pglatex. %% +%% Compile this project with: %% +%% pdflatex 37354-t.tex ..... TWO times %% +%% %% +%% pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) %% +%% %% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\listfiles + +\documentclass[12pt]{book}[2007/10/19] +\usepackage[latin1]{inputenc}[2008/03/30] + +\usepackage{amsmath}[2000/07/18] %% Displayed equations +\usepackage{amssymb}[2009/06/22] %% and additional symbols + +\usepackage{alltt}[1997/06/16] %% boilerplate, credits, license + +\usepackage{array}[2008/09/09] %% extended array/tabular features + +\usepackage{ifthen}[2001/05/26] + +% for running heads +\usepackage{fancyhdr} + +\newcommand{\Margins}{hmarginratio=1:1} % Symmetric margins +\newcommand{\HLinkColor}{blue} % Hyperlink color +\newcommand{\PDFPageLayout}{SinglePage} +\newcommand{\TransNote}{Transcriber's Note} +\newcommand{\TransNoteText}{% + Minor typographical corrections and presentational changes have + been made without comment. + \bigskip +} + +\renewcommand{\cleardoublepage}{\clearpage} + + \raggedbottom + \usepackage[body={5in,6.66in}, paperwidth=5.25in, paperheight=8in, \Margins,includeheadfoot]{geometry}[2010/09/12] + +\providecommand{\ebook}{00000} % Overridden during white-washing +\usepackage[pdftex, + hyperfootnotes=false, + pdftitle={The Project Gutenberg eBook \#\ebook: Condensation of Determinants, being a new and brief Method for computing their arithmetical values}, + pdfauthor={Charles L. Dodgson (Lewis Carrol)}, + pdfkeywords={Carlo Traverso, Anna Hall, + Project Gutenberg Online Distributed Proofreading Team}, + pdfstartview=Fit, % default value + pdfstartpage=1, % default value + pdfpagemode=UseNone, % default value + bookmarks=true, % default value + linktocpage=false, % default value + pdfpagelayout=\PDFPageLayout, + pdfdisplaydoctitle, + pdfpagelabels=true, + bookmarksopen=true, + bookmarksopenlevel=1, + colorlinks=true, + linkcolor=\HLinkColor]{hyperref}[2011/04/17] + +%%%% Fixed-width environment to format PG boilerplate %%%% +\newenvironment{PGtext}{% +\begin{alltt} +\fontsize{8.1}{9}\ttfamily\selectfont}% +{\end{alltt}} + +% Running heads +\newcommand{\FlushRunningHeads}{\clearpage\fancyhf{}\cleardoublepage} + +\newcommand{\BookMark}[2]{\phantomsection\pdfbookmark[#1]{#2}{#2}} + +%% Major document divisions %% +\newcommand{\PGBoilerPlate}{% + \pagenumbering{Alph} + \pagestyle{empty} + \BookMark{0}{PG Boilerplate.} +} +\newcommand{\FrontMatter}{% + \cleardoublepage + \frontmatter + \pagestyle{empty} +} + +\newcommand{\MainMatter}{% + \FlushRunningHeads + \mainmatter + \pagestyle{fancy} + \fancyhf{} + \setlength{\headheight}{15pt} + \thispagestyle{plain} + \fancyhead[C]{Rev.\ C. L. Dodgson \textit{on Condensation of Determinants}.} + \renewcommand{\headrulewidth}{0pt} + \fancyhead[R]{\thepage} +} + +\newcommand{\BackMatter}{% + \FlushRunningHeads + \backmatter + \BookMark{-1}{Back Matter.} +} + +\newcommand{\DPtypo}[2]{#2} + +\newcommand{\SquashArrays}{\setlength{\arraycolsep}{2pt}} + +\newcommand{\tb}{\begin{center}\rule{1in}{0.5pt}\end{center}} + +\DeclareInputText{183}{\ifmmode\cdot\else\textperiodcentered\fi} + +\newcounter{RowCount} +% \Tall{3}: Create vertical space as tall as an array with 3 rows +\newcommand{\Tall}[1]{\setcounter{RowCount}{1}% + \vphantom{\left|\begin{array}{r} + \whiledo{\value{RowCount} < #1} + {\stepcounter{RowCount}0\\} + 0 + \end{array}\right|}} + +\hfuzz=0.5pt +%%%%%%%%%%%%%%%%%%%%%%%% START OF DOCUMENT %%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{document} +\PGBoilerPlate +{\small +\begin{PGtext} +The Project Gutenberg EBook of Condensation of Determinants, Being a New +and Brief Method for Computing their Arithmetical Values, by Lewis Carroll (AKA Charles Lutwidge Dodgson) + +This eBook is for the use of anyone anywhere at no cost and with +almost no restrictions whatsoever. You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org + + +Title: Condensation of Determinants, Being a New and Brief Method for Computing their Arithmetical Values + +Author: Lewis Carroll (AKA Charles Lutwidge Dodgson) + +Release Date: September 8, 2011 [EBook #37354] + +Language: English + +Character set encoding: ISO-8859-1 + +*** START OF THIS PROJECT GUTENBERG EBOOK CONDENSATION OF DETERMINANTS *** +\end{PGtext} +\clearpage +\begin{PGtext} +Produced by Carlo Traverso, Anna Hall and the Online +Distributed Proofreading Team at http://www.pgdp.net (This +book was produced from scanned images of public domain +material from the Google Print project.) +\end{PGtext}} +\vfill + +\BookMark{0}{Transcriber's Note} +\subsection*{\centering\normalfont\scshape +\normalsize\MakeLowercase{\TransNote}} + +\TransNoteText +\clearpage + +%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%% + +\FrontMatter + +%% -----File: 000.png--- +\begin{center} +{\LARGE PROCEEDINGS} \\ +\vfill +{\scriptsize OF THE }\\ +\vfill +{\LARGE ROYAL SOCIETY OF LONDON.} +\vfill +\textit{From January~11, 1866, to May~23, 1867, inclusive.} +\vfill +VOL.~XV. +\vfill +LONDON: \\ +PRINTED BY TAYLOR AND FRANCIS, \\ +{\scriptsize RED LION COURT, FLEET STREET.} \\ +{\footnotesize MDCCCLXVII.} +\end{center} +%% -----File: 001.png--- +\MainMatter +\begin{quote} +IV. ``Condensation of Determinants, being a new and brief Method +for computing their arithmetical values.'' By the Rev.\ C.~L. +\textsc{Dodgson},~M.A., Student of Christ Church, Oxford. Communicated +by the Rev.\ \textsc{Bartholomew Price}, M.A., F.R.S\@. +Received May~15,~1866. +\end{quote} + +If it be proposed to solve a set of $n$~simultaneous linear equations, not +being all homogeneous, involving $n$~unknowns, or to test their compatibility +when all are homogeneous, by the method of determinants, in these, as +well as in other cases of common occurrence, it is necessary to compute +the arithmetical values of one or more determinants---such, for example, as +\[ +\left|\begin{array}{rrr} +1, &3, &-2\\ +2, &1, & 4\\ +3, &5, &-1 +\end{array}\right|. +\] + +Now the only method, so far as I am aware, that has been hitherto +employed for such a purpose, is that of multiplying each term of the first +row or column by the determinant of its complemental minor, and affecting +the products with the signs $+$~and~$-$ alternately, the determinants required +in the process being, in their turn, broken up in the same manner +until determinants are finally arrived at sufficiently small for mental computation. + +This process, in the above instance, would run thus:--- +\[ +\begin{split} +\left|\begin{array}{rrr} +1, &3, &-2\\ +2, &1, & 4\\ +3, &5, &-1 +\end{array}\right| += 1 × \left|\begin{array}{rr} +1, & 4\\ +5, &-1 +\end{array}\right| +-2 × \left|\begin{array}{rr} +3, &-2\\ +5, &-1 +\end{array}\right| ++ 3 × \left|\begin{array}{rr} +3, &-2\\ +1, & 4 +\end{array}\right|\\ += -21 - 14 + 42 = 7. +\end{split} +\] + +But such a process, when the block consists of $16$, $25$,~or more terms, is +so tedious that the old method of elimination is much to be preferred for +solving simultaneous equations; so that the new method, excepting for +equations containing $2$~or $3$~unknowns, is practically useless. + +The new method of computation, which I now proceed to explain, and +for which ``Condensation'' appears to be an appropriate name, will be +found, I believe, to be far shorter and simpler than any hitherto employed. + +In the following remarks I shall use the word ``Block'' to denote any +number of terms arranged in rows and columns, and ``interior of a block'' +to denote the block which remains when the first and last rows and columns +are erased. + +The process of ``Condensation'' is exhibited in the following rules, in +which the given block is supposed to consist of $n$~rows and $n$~columns:--- + +(1) Arrange the given block, if necessary, so that no ciphers occur in its +interior. This may be done either by transposing rows or columns, or by +adding to certain rows the several terms of other rows multiplied by +certain multipliers. + +(2) Compute the determinant of every minor consisting of four adjacent +%% -----File: 002.png--- +terms. These values will constitute a second block, consisting of $(n - 1)$ +rows and $(n - 1)$~columns. + +(3) Condense this second block in the same manner, dividing each term, +when found, by the corresponding term in the interior of the first block. + +(4) Repeat this process as often as may be necessary (observing that in +condensing any block of the series, the $r$th for example, the terms so found +must be divided by the corresponding terms in the interior of the $(r - 1)$th +block), until the block is condensed to a single term, which will be the +required value. + +As an instance of the foregoing rules, let us take the block +\[ +\left|\begin{array}{rrrr} +-2 &-1 &-1 &-4\\ +-1 &-2 &-1 &-6\\ +-1 &-1 & 2 & 4\\ + 2 & 1 &-3 &-8 +\end{array}\right|. +\] + +By rule~(2) this is condensed into +$\left|\begin{array}{rrr} + 3 &-1 & 2\\ +-1 &-5 & 8\\ + 1 & 1 &-4 +\end{array}\right| +$; +this, again, by +rule~(3), is condensed into +$\left|\begin{array}{rr} + 8 &-2\\ +-4 & 6 +\end{array}\right|$; and this, by rule~(4), into~$-8$, +which is the required value. + +The simplest method of working this rule appears to be to arrange the +series of blocks one under another, as here exhibited; it will then be found +very easy to pick out the divisors required in rules (3)~and~(4). +\[ +\left|\begin{array}{rrrr} +-2 &-1 &-1 &-4\\ +-1 &-2 &-1 &-6\\ +-1 &-1 & 2 & 4\\ + 2 & 1 &-3 &-8 +\end{array}\right| +\] +\[ +\left|\begin{array}{rrr} + 3 &-1 &2\\ +-1 &-5 &8\\ + 1 & 1 &-4 +\end{array}\right| +\] +\[ +\left|\begin{array}{rr} + 8 &-2\\ +-4 & 6 +\end{array}\right| +\] +\[ +-8. +\] + +This process cannot be continued when ciphers occur in the interior of +any one of the blocks, since infinite values would be introduced by employing +them as divisors. When they occur in the given block itself, it +may be rearranged as has been already mentioned; but this cannot be done +when they occur in any one of the derived blocks; in such a case the +given block must be rearranged as circumstances require, and the operation +commenced anew. + +The best way of doing this is as follows:--- + +Suppose a cipher to occur in the $h$th~row and $k$th~column of one of the +derived blocks (reckoning both row and column from the \emph{nearest} corner +of the block); find the term in the $h$th~row and $k$th~column of the given +%% -----File: 003.png--- +block (reckoning from the corresponding corner), and transpose rows or +columns cyclically until it is left in an outside row or column. When the +necessary alterations have been made in the derived blocks, it will be found +that the cipher now occurs in an outside row or column, and therefore +need no longer be used as a divisor. + +The advantage of \emph{cyclical} transposition is, that most of the terms in the +new blocks will have been computed already, and need only be copied; in +no case will it be necessary to compute more than \emph{one} new row or column +for each block of the series. + +In the following instance it will be seen that in the first series of blocks +a cipher occurs in the interior of the third. We therefore abandon the +process at that point and begin again, rearranging the given block by +transferring the top row to the bottom; and the cipher, when it occurs, is +now found in an exterior row. It will be observed that in each block of +the new series, there is only \emph{one} new row to be computed; the other rows +are simply copied from the work already done. +\[ +\begin{gathered}[t] +\left|\begin{array}{rrrrr} +2 &-1 & 2 & 1 &-3\\ +1 & 2 & 1 &-1 & 2\\ +1 &-1 &-2 &-1 &-1\\ +2 & 1 &-1 &-2 &-1\\ +1 &-2 &-1 &-1 & 2\\ +\end{array}\right| +\\ +\left|\begin{array}{rrrr} + 5 &-5 &-3 &-1\\ +-3 &-3 &-3 & 3\\ + 3 & 3 & 3 &-1\\ +-5 &-3 &-1 &-5\\ +\end{array}\right| +\\ +\left|\begin{array}{rrr} +\DPtypo{-30}{-15} & 6 &\DPtypo{-12}{12}\\ + 0 & 0 & 6\\ + 6 &-6 & 8\\ +\end{array}\right| +\end{gathered} +\qquad +\begin{gathered}[t] +\left|\begin{array}{rrrrr} +1 & 2 & 1 &-1 & 2\\ +1 &-1 &-2 &-1 &-1\\ +2 & 1 &-1 &-2 &-1\\ +1 &-2 &-1 &-1 & 2\\ +2 &-1 & 2 & 1 &-3\\ +\end{array}\right| +\\ +\left|\begin{array}{rrrr} +-3 &-3 &-3 & 3\\ + 3 & 3 & 3 &-1\\ +-5 &-3 &-1 &-5\\ + 3 &-5 & 1 & 1\\ +\end{array}\right| +\\ +\left|\begin{array}{rrr} + 0 & 0 & 6\\ + 6 &-6 & 8\\ +-17 & 8 &-4\\ +\end{array}\right| +\\ +\left|\begin{array}{rr} + 0 &12\\ +18 &40\\ +\end{array}\right| +\\ +36. +\end{gathered} +\] + +The fact that, whenever ciphers occur in the interior of a derived block, +it is necessary to recommence the operation, may be thought a great +obstacle to the use of this method; but I believe it will be found in practice +that, even though this should occur several times in the course of one +operation, the whole amount of labour will still be much less than that involved +in the old process of computation. +\tb + +I now proceed to give a proof of the validity of this process, deduced +from a well-known theorem in determinants; and in doing so, I shall use +the word ``adjugate'' in the following sense:---if there be a square block, +and if a new block be formed, such that each of its terms is the determinant +of the complemental minor of the corresponding term of the first +block, the second block is said to be \emph{adjugate} to the first. +%% -----File: 004.png--- + +The theorem referred to is the following:--- + +``If the determinant of a block $= R$, the determinant of any minor of +the $m$th~degree of the adjugate block is the product of~$R^{m - 1}$ and the +coefficient which, in~$R$, multiplies the determinant of the corresponding +minor.'' + +Let us first take a block of $9$~terms, +\[ +\left|\begin{array}{ccc} +a_{1,1} &a_{1,2} &a_{1,3}\\ +a_{2,1} &a_{2,2} &a_{2,3}\\ +a_{3,1} &a_{3,2} &a_{3,3} +\end{array}\right| += R; +\] +and let $\alpha_{1,1}$~represent the determinant of the complemental minor of~$a_{1,1}$, +and so on. + +If we ``condense'' this, by the method already given, we get the block +$\left\{\begin{array}{cc} +\alpha_{3,3} &\alpha_{3,1}\\ +\alpha_{1,3} &\alpha_{1,1} +\end{array}\right\},$ and, by the theorem above cited, the determinant of this, +viz.\ +\[ +\left|\begin{array}{cc} +\alpha_{3,3} &\alpha_{3,1}\\ +\alpha_{1,3} &\alpha_{1,1} +\end{array}\right| += R × a_{2,2}. +\] + +Hence +\[ +R = \frac{\left|\begin{array}{cc} +\alpha_{3,3} &\alpha_{3,1}\\ +\alpha_{1,3} &\alpha_{1,1} +\end{array}\right|}{a_{2,2}}, +\] +which proves the rule. + +Secondly, let us take a block of $16$~terms: +\[ +\left|\begin{array}{ccc} +a_{1,1}&\dots &a_{1,4}\\ +\vdots & &\vdots \\ +a_{4,1}&\dots &a_{4,4} +\end{array}\right| += R. +\] +If we ``condense'' this, we get a block of $9$~terms; let us denote it by +\[ +\left\{\begin{array}{ccc} +b_{1,1} &\dots &b_{1,3}\\ +\vdots & &\vdots \\ +b_{3,1} &\dots &b_{3,3} +\end{array}\right\}, +\text{ in which } +b_{1,1} = +\left|\begin{array}{cc} +a_{1,1} &a_{1,2}\\ +a_{2,1} &a_{2,2} +\end{array}\right|, \text{ \&c.} +\] + +If we ``condense'' this block again, we get a block of $4$~terms, each +of which, by the preceding paragraph, is the determinant of $9$~terms +of the original block; that is to say, we get the block +$\left\{\begin{array}{cc} +\alpha_{4,4} &\alpha_{4,1}\\ +\alpha_{1,4} &\alpha_{1,1} +\end{array}\right\}$; +but, by the theorem already quoted, +$\left|\begin{array}{cc} +\alpha_{4,4} &\alpha_{4,1}\\ +\alpha_{1,4} &\alpha_{1,1} +\end{array}\right| = R × b_{2,2}$; therefore +$R = \dfrac{\left|\begin{array}{cc} +\alpha_{4,4} &\alpha_{4,1}\\ +\alpha_{1,4} &\alpha_{1,1} +\end{array}\right|}{b_{2,2}}$; that is, $R$~may be obtained by ``condensing'' the block +$\left\{\begin{array}{cc} +\alpha_{4,4} &\alpha_{4,1}\\ +\alpha_{1,4} &\alpha_{1,1} +\end{array}\right\}$. + +This proves the rule for a block of $16$~terms; and similar proofs might +be given for larger blocks. +\tb + +I shall conclude by showing how this process may be applied to the +solution of simultaneous linear equations. +%% -----File: 005.png--- + +If we take a block consisting of $n$~rows and $(n + 1)$~columns, and ``condense'' +it, we reduce it at last to $2$~terms, the first of which is the determinant +of the first $n$~columns, the other of the last $n$~columns. + +Hence, if we take the $n$~simultaneous equations, +\begin{alignat*}{3} +a_{1,1}x_{1} +{}& a_{1,2}x_{2} + \cdots\,&\cdots + a_{1,n}x_{n} &+ a_{1,n + 1} = 0,\\ +\multispan{4}{\dotfill} \\ +a_{n,1}x_{1} +{}&\multispan{2}{\dotfill}&+ a_{n,n + 1} = 0; +\end{alignat*} +and if we condense the whole block of coefficients and constants, viz.\ +\[ +\left\{\begin{array}{ccc} +a_{1,1} &\dots &a_{1,n + 1}\\ +\vdots & &\vdots \\ +a_{n,1} &\dots &a_{n,n + 1} +\end{array}\right\}, +\] +we reduce it at last to $2$~terms: let us denote them by $S$,~$T$, so that +\[ +S = \left|\begin{array}{ccc} +a_{1,1} &\dots &a_{1,n}\\ +\vdots & &\vdots \\ +a_{n,1} &\dots &a_{n,n} +\end{array}\right|, +\text{ and } +T = \left|\begin{array}{ccc} +a_{\DPtypo{1,1}{1,2}} &\dots &a_{1,n+1}\\ +\vdots & &\vdots \\ +a_{n,2} &\dots &a_{n,n+1} +\end{array}\right|. +\] + +Now we know that $x_{1} = (-)^{n} \dfrac{T}{S}$, which may be written in the form +$(-)^{n}S · x_{1} = T$. + +Hence the $2$~terms obtained by the process of condensation may be +converted into an equation for~$x_{1}$, by multiplying the first of them by~$x_{1}$, +affected with $+$~or~$-$, according as $n$~is even or odd. The latter part of +the rule may be simply expressed thus:---``place the signs $+$~and~$-$ +alternately over the several columns, beginning with the last, and the sign +which occurs over the column containing~$x_{1}$ is the sign with which $x_{1}$ is to +be affected.'' + +When the value of~$x_{1}$ has been thus found, it may be substituted in the +first $(n - 1)$~equations, and the same operation repeated on the new block, +which will now consist of $(n - 1)$~rows and $n$~columns. But in calculating +the second series of blocks, it will be found that most of the work has been +already done; in fact, of the $2$~determinants required in the new block, one +has been already computed correctly, and the other so nearly so that it +only requires the \emph{last} column in each of the derived blocks to be corrected. + +In the \hyperlink{example}{example} given opposite, after writing $+$~and~$-$ alternately over +the columns, beginning with the last, we first condense the whole block, and +thus obtain the $2$~terms $36$~and~$-72$. Observing that the $x$-column has +the sign~$-$ placed over it, we multiply the~$36$ by~$-x$, and so form the +equation $-36x = -72$, which gives~$x = 2$. + +Hence the $x$-terms in the first four equations become respectively +$2$,~$2$,~$4$, and~$2$; adding these values to the constant terms in the same equations, +we obtain a block of which we need only write down the last two +columns, viz.\ +\[ +\left|\begin{array}{rr} + 2 & 4\\ +-1 &-2\\ +-1 &-2\\ + 2 & 6 +\end{array}\right|. +\] +%% -----File: 006.png--- + +We then condense these into the column~$\left|\begin{array}{r} +0\\ +0\\ +\DPtypo{2}{-2} +\end{array}\right|$, +and, supplying from +the second block of the first series the column~$\left|\begin{array}{r} + 3\\ +-1\\ +-5 +\end{array}\right|$, we obtain +$\left|\begin{array}{rr} + 3 &0\\ +-1 &0\\ +-5 &\DPtypo{2}{-2} +\end{array}\right|$ as the last two columns of the \emph{second} block of the new series; +and proceeding thus we ultimately obtain the two terms $12$,~$12$. Observing +that the $y$-column has the sign~$+$ placed over it, we multiply the first~$12$ +by~$+y$, and so form the equation $12y = 12$, which gives $y = 1$. The +values of $z$,~$u$, and~$v$ are similarly found. + +% In Dodgson's two examples, the coefficient matrix of the original +% system is at upper left. Each column of the calculation, which Dodgson +% calls a "series", solves the system for one variable. Successive matrices +% in a column are obtained by condensations, reducing the size of the matrix +% until the value of one variable can be read off, with the sign adjusted +% according to the +/- over the corresponding column in the original system. +% +% Next, the solved value of the variable is substituted into the system, the +% last equation/coefficient row is dropped, the constants are "updated", and +% the process begins anew. Only condensations involving the last column need +% to be recomputed, so the second and subsequent series involve only 2 x n +% matrices. In the matrix at the top of each series, the two columns are +% (i) the penultimate column of the preceding series and (ii) the updated +% values of the constants. +It will be seen that when once the given block has been successfully +condensed, and the value of the first unknown obtained, there is no further +danger of the operation being interrupted by the occurrence of ciphers. +\hypertarget{example} +\[ +\begin{array}{ccccccc} + - &+ &- &+ &- &+ & \\ + x &+2y &+ z &- u &+2v &+2 &= 0\\ + x &- y &-2z &- u &- v &-4 &= 0\\ +2x &+ y &- z &-2u &- v &-6 &= 0\\ + x &-2y &- z &- u &+2v &+4 &= 0\\ +2x &- y &+2z &+ u &-3v &-8 &= 0 +\end{array} +\] +\[ +\SquashArrays +% First series of blocks +\begin{gathered} +\left|\begin{array}{rrrrrr} +1 & 2 & 1 &-1 & 2 & 2\\ +1 &-1 &-2 &-1 &-1 &-4\\ +2 & 1 &-1 &-2 &-1 &-6\\ +1 &-2 &-1 &-1 & 2 & 4\\ +2 &-1 & 2 & 1 &-3 &-8 +\end{array}\right| +\\ +\left|\begin{array}{rrrrr} +-3 &-3 &-3 & 3 &-6\\ + 3 & 3 & 3 &-1 & 2\\ +-5 &-3 &-1 &-5 & 8\\ + 3 &-5 & 1 & 1 &-4 +\end{array}\right| +\\ +\left|\begin{array}{rrrr} + 0 & 0 & 6 & 0\\ + 6 &-6 & 8 &-2\\ +-17 & 8 &-4 & 6 +\end{array}\right| +\\ +\left|\begin{array}{rrr} + 0 &12 &12\\ +18 &40 &-8 +\end{array}\right| +\\ +\left|\begin{array}{rr} 36 & -72 \end{array}\right| \\ +\DPtypo{\because}{\therefore} -36x = -72 \\ +x = 2 +\end{gathered} +\; +% Second series +\begin{gathered} +\left|\begin{array}{rr} + 2 & 4\\ +-1 &-2\\ +-1 &-2\\ + 2 & 6 +\end{array}\right| +\\ +\left|\begin{array}{rr} + 3 & 0\\ +-1 & 0\\ +-5 &-2 +\end{array}\right| +\\ +\left|\begin{array}{rr} +6 & 0\\ +8 &-2 +\end{array}\right| +\\ +\left|\begin{array}{rr} 12 & 12 \end{array}\right| \\ +\therefore 12y = 12 \\ +y = 1 +\\ +\Tall{5} +\end{gathered} +\; +% Third series +\begin{gathered} +\left|\begin{array}{rr} + 2 & 6\\ +-1 &-3\\ +-1 &-1 +\end{array}\right| +\\ +\left|\begin{array}{rr} + 3 & 0\\ +-1 &-2 +\end{array}\right| +\\ +\left|\begin{array}{rr} 6 & 6\end{array}\right| \\ +\therefore -6z = 6 \\ +z = -1 +\\ +\Tall{5} +\\ +\Tall{4} +\end{gathered} +\; +% Fourth series +\begin{gathered} +\left|\begin{array}{rr} + 2 & 5\\ +-1 &-1 +\end{array}\right| +\\ +\left|\begin{array}{rr} 3 & 3 \end{array}\right| \\ +\therefore 3u = 3 \\ +u = 1 +\\ +\Tall{5} +\\ +\Tall{4} +\\ +\Tall{3} +\end{gathered} +\; +% Fifth series +\begin{gathered} +\left|\begin{array}{rr} 2 & 4 \end{array}\right| \\ +\therefore -2v = 4 \\ +v = -2 +\\ +\Tall{5} +\\ +\Tall{4} +\\ +\Tall{3} +\\ +\Tall{2} +\end{gathered} +\] + +\tb +% +\[ +\begin{array}{ccccc} +- &+ &- &+ & \\ +5x &+2y &-3z &+ 3 &= 0\\ +3x &- y &-2z &+ 7 &= 0\\ +2x &+3y &+ z &-12 &= 0 +\end{array} +\] +\[ +% First series +\begin{gathered} +\left|\begin{array}{rrrr} +5 & 2 &-3 & 3\\ +3 &-1 &-2 & 7\\ +2 & 3 & 1 &-12 +\end{array}\right| +\\ +\left|\begin{array}{rrr} +-11 &-7 &-15\\ + 11 & 5 & 17 +\end{array}\right| +\\ +\left|\begin{array}{cc} -22 & 22 \end{array}\right| \\ +\therefore 22x =22 \\ +x = 1 +\end{gathered} +\quad +\begin{gathered} +\left|\begin{array}{rr} +-3 & 8\\ +-2 &10 +\end{array}\right| +\\ +\left|\begin{array}{cc} -7 & -14 \end{array}\right| \\ +\therefore -7y = -14 \\ +y = 2 +\\ +\Tall{3} +\end{gathered} +\quad +\begin{gathered} +\left|\begin{array}{rr} -3 & 12 \end{array}\right| \\ +\therefore 3z = 12 \\ +z = 4 +\\ +\Tall{3} +\\ +\Tall{2} +\end{gathered} +\] + +The Society then adjourned over the Whitsuntide Recess to Thursday, +May~31. + +\BackMatter +%%%%%%%%%%%%%%%%%%%%%%%%% GUTENBERG LICENSE %%%%%%%%%%%%%%%%%%%%%%%%%% +\newpage +\BookMark{0}{PG License.} +\begin{PGtext} +End of the Project Gutenberg EBook of Condensation of Determinants, Being a +New and Brief Method for Computing their Arithmetical Values, by Lewis Carroll (AKA Charles Lutwidge Dodgson) + +*** END OF THIS PROJECT GUTENBERG EBOOK CONDENSATION OF DETERMINANTS *** + +***** This file should be named 37354-pdf.pdf or 37354-pdf.zip ***** +This and all associated files of various formats will be found in: + http://www.gutenberg.org/3/7/3/5/37354/ + +Produced by Carlo Traverso, Anna Hall and the Online +Distributed Proofreading Team at http://www.pgdp.net (This +book was produced from scanned images of public domain +material from the Google Print project.) + + +Updated editions will replace the previous one--the old editions +will be renamed. + +Creating the works from public domain print editions means that no +one owns a United States copyright in these works, so the Foundation +(and you!) can copy and distribute it in the United States without +permission and without paying copyright royalties. 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in +jurisdictions other than the United States. Anyone seeking to utilize +this eBook outside of the United States should confirm copyright +status under the laws that apply to them. diff --git a/README.md b/README.md new file mode 100644 index 0000000..f1795a2 --- /dev/null +++ b/README.md @@ -0,0 +1,2 @@ +Project Gutenberg (https://www.gutenberg.org) public repository for +eBook #37354 (https://www.gutenberg.org/ebooks/37354) |