The (almost really) Complete Works of Lewis Carroll

with Their Application to Simultaneous Linear Equations and Algebraical Geometry

Preface

Of the seventy Propositions contained in the following treatise, ten are substantially taken from Baltzer’s treatise on Determinants; also the Geometrical Tests, given in Chapter VIII, are to be found in most works on Algebraical Geometry: the rest of the matter is, so far as I know, original, and consists of a series of Propositions which the object I had in view obliged me to introduce. That object was to present the subject as a continuous chain of argument, separated from all accessories of explanation or illustration, a form which I venture to think better suited for a treatise on exact science than the semi-colloquial semi-logical form often adopted by Mathematical writers. I say ‘semi-logical’ advisedly, for nothing is more easy than to forget, in an argument thus interwoven with illustrative matter, what has, and what has not, been proved.

With this object in view I have introduced all such explanation and illustration as seemed necessary for a beginner, either in the form of footnotes, or, where that would have occupied too much room, of Appendices.

New words and symbols are always a most unwelcome addition to a Science, especially to one already burdened with an enormous vocabulary, yet I think the Definitions given of them will be found to justify their introduction, as the only way of avoiding tedious periphrasis. The symbols employed to represent the single elements of a Determinant, ($12$, $13$, &c.) require perhaps a word of apology, and it may be well to enumerate those already in use, and to point out what seem to be their chief defects.

We may commence with $\left\{\begin{array}{c}{a}_1,b_1,\dots \\ a_2,b_2,\dots \end{array}\right\}$, where the change of letter indicates a change of column, and the change of subscript a change of row. Now the properties of Determinants, relating to columns, being always convertible into properties relating to rows, and vice versa, it was a sufficient objection to this system of notation, that it represented things distinctly analogous by methods so different, and it was properly superseded by the notation introduced by Leibnitz, $\left\{\begin{array}{c}{a}_{1,1},a_{1,2},\dots \\ a_{2,1},a_{2,2},\dots \end{array}\right\}$, where the changes, both of column and row, are alike denoted by subscripts. But it seems a fatal objection to this system that most of the space is occupied by a number of a’s, which are wholly superfluous, while the only important part of the notation is reduced to minute subscripts, alike difficult to the writer and the reader. It was almost an obvious improvement on this system to raise the subscripts into the line, and omit the a’s altogether, as suggested by Baltzer, thus— $\left\{\begin{array}{c}(1,1),(1,2),\dots \\ (2,1),(2,2),\dots \end{array}\right\}$ and this system, though tedious for writing, might serve very well, were it not for its liability to be confused with the notation, common in Plane Algebraical Geometry, by which $(1,1)$ denotes the Point $x=1$, $y=1$. The symbol $11$, which I have ventured to suggest as an emendation on this last, will be found, I have great hopes, sufficiently simple, distinct, and easy to be written. I have turned the symbol towards the left, in order to avoid all chance of confusion with ∫, the symbol for integration.

I proceed to make a few introductory remarks on the various portions of the book, taken in order.

Chap. II. Def. I. I am aware that the word ‘Matrix’ is already in use to express the very meaning for which I use the word ‘Block’; but surely the former word means rather the mould, or form, into which algebraical quantities may be introduced, than an actual assemblage of such quantities; for instance, $\frac{(\phantom{a+b})\times (\phantom{c+d})}{(\phantom{e+f})}$ would deserve the name, rather than $\frac{(a+b)\times (c+d)}{(e+f)}$.

Chap. II. Def. I, VIII. Those who have read the chapters on Determinants in Mr. Todhunter’s ‘Theory of Equations’ will notice that the meanings of the words ‘Element’ and ‘Constituent’ are here transposed: as to the former, I have only returned to Baltzer’s nomenclature; and the word ‘Constituent’ seems to me more expressive than his word ‘Term’.

Chap. III. A complete analysis of a system of simultaneous Linear Equations has always appeared to me to be a desideratum in Algebra: the subject is only touched on in Baltzer; a more complete attempt will be found in Peacock’s Algebra, but I have nowhere seen anything like an exhaustive analysis. This chapter aims at furnishing this, but it has been so often altered and re-written that I put it forth at last, hoping, rather than expecting, that it will be found complete and satisfactory.

Chap. VII. This chapter will also, I hope, fulfil my aim at furnishing an exhaustive analysis of such properties of the Loci here considered, as can be conveniently exhibited in the form of Determinants. I had added propositions concerning the Line in Solid Geometry, but these I omit, believing that its properties are more simply investigated by other methods.

Appendix II. Section 4. This process, though extremely convenient where no ciphers, or where one or two at most, occur in the interior of a Block, nevertheless fails entirely, it must be admitted, where they occur in larger numbers: I therefore offer it merely as a fanciful addition to the processes already in use, which may in some cases lessen the labour of computation.

Appendix V. I am doubtful whether this process will ever prove of much practical use: still I think cases might arise, where in the course of a problem an algebraical function is proved to vanish, and where, by throwing it into the form of a Determinant, and so forming a set of simultaneous Equations, whose consistency depends on its vanishing, new and curious properties of the function under consideration might be evolved.

The formulæ given at the end of the book are so arranged that the student may, by covering one or more of the columns on the right hand, test for himself his knowledge of the theorems from which they are taken.

Chapter I. Laws of Arrangement

Definitions

I.

A set of different numerals, arranged in an ascending order, is said to be orderly arranged: but if there be among them 2, of which the second is less than the first, the set is said to contain a derangement.¹

II.

If 2 numbers be both even, or both odd, they are said to be similar; if otherwise, dissimilar.

Proposition I. Th.

If there be a set of different numerals, arranged in any order, and if one of them be made to pass over the next r of them, either way: the number of derangements is increased, or diminished, by a number similar to r.

If it be made to pass over one, the number is increased or diminished, by unity;

∴ if over two, by an even number;

∴ if over three, by an odd number; and so on.

Therefore, if there be a set, &c. Q. E. D.²

Proposition II. Th.

If there be a set of different numerals, and if 2 of them be interchanged: the number of derangements is increased, or diminished, by an odd number.

Call the 2 numerals, α, β; and let there be r numerals between them;

firstly, let α be made to pass over these r numerals;

then the number of derangements is increased, or diminished, by a number similar to r; (Prop. I.

secondly, let β be made to pass over α and over these r numerals;

then the number of derangements is thereby increased, or diminished, by a number similar to $\overline{r+1}$; (Prop. I.

∴ it is ultimately increased, or diminished, by the sum or difference of 2 dissimilar numbers;

i. e. it is increased, or diminished, by an odd number.

Therefore, if there be a set, &c. Q. E. D.³

Proposition III. Th.

If there be a set of pairs of numerals, in which the antecedents are all different, as also are the consequents; and if they be arranged, firstly in order of antecedents, and secondly in order of consequents: the number of derangements among the consequents in the first case, and the number of derangements among the antecedents in the second case, are equal.

Let the pairs be so placed that the antecedents are orderly arranged, and let 2 of them be selected, and call them $(H,r)$, $(K,s)$;

$\therefore H<K$;

now, if these 2 pairs contain a derangement of consequents, $r>s$;

∴ when the pairs are re-arranged in order of consequents, these 2 will stand in the order … $(K,s)$ … $(H,r)$ …;

∴ they will then contain a derangement of antecedents;

but if these 2 pairs do not contain a derangement of consequents, $r<s$;

∴ when the pairs are re-arranged in order of consequents, these 2 will stand in the order … $(H,r)$ … $(K,s)$ …;

∴ they will then not contain a derangement of antecedents.

And the same thing may be proved for every other 2 pairs.

Therefore, if there be, &c. Q. E. D.⁴

Definition III.

If there be a set of pairs of numerals, in which the antecedents are all different, as also are the consequents; and if, when they are arranged in order of antecedents, the number of derangements among the consequents be even, or (which is the same thing) if, when they are arranged in order of consequents, the number of derangements among the antecedents be even: the set is said to be of the even class; if otherwise, of the uneven class.⁵

Proposition IV. Th.

If there be a set of pairs of uumerals, in which the antecedents are all different, as also are the consequents; and if 2 of the antecedents, or 2 of the consequents, be interchanged: the class, to which the set belongs, is changed.

Let the set be arranged iu order of antecedents, and let 2 of the consequents be interchanged;

then the number of derangements among them is increased, or diminished, by an odd number; (Prop. II.

i. e. if even, it becomes odd; if odd, even;

∴ the class, to which the set belongs, is changed;

hence, if the set be arranged in any order, and 3 of the consequents be interchanged, the class, to which the set belongs, is changed.

Similarly, if 2 of the antecedents be interchanged.

Therefore, if there be, &c. Q. E. D.⁶

Proposition V. Th.

If there be a set of n pairs of numerals, in which the antecedents are a certam permutation of the numbers from 1 to n, as also are the consequents; and if one pair be erased: the class, to which the remaining set belongs, is the same as that of the original set, or different, accordmg as the numerals in the erased pair are similar or dissimilar.

Let the set be arranged in order of antecedents; and call the pair that is to be erased $(H,k)$;

firstly, let it be brought to the first place, by making it pass over the preceding $\overline{H-1}$ pairs;

then the number of derangements among the consequents is increased, or diminished, by a number similar to $\overline{H-1}$; (Prop. I.

and, since the consequent k now precedes the $\overline{k-1}$ consequents which are less than it, there are now, by reason of this pair, $\overline{k-1}$ derangements among the consequents;

secondly, let the pair $(H,k)$ be erased;

then the number of derangements among the consequents is thereby diminished by a number similar to $\overline{k-1}$;

∴ it is ultimately increased, or diminished, by a number similar to the sum or difference of $\overline{H-1}$ and $\overline{k-1}$;

i. e. it is ultimately increased, or diminished, by an even, or odd, number, according as $\overline{H-1}$ and $\overline{k-1}$ are similar or dissimilar;

i. e. according as H and k are similar or dissimilar.

Therefore, if there be a set, &c. Q. E. D.⁷.

Chapter II. Analysis of Determinants

Definitions.

I.

If $mn$ quantities be so placed as to form m rows and n columns: they are said to form a Block; and the $mn$ quantities are called the Elements of such a Block.

II.

A square Block of $n^2$ Elements is said to be of the n^th degree.

III.

An oblong Block containing m rows and n columns, or m columns and n rows, where m is greater than n, is said to be of the length m, and of the breadth n.

IV.

In an oblong Block, the rows, if they be longer than the columns, or the columns, if they be longer than the rows, are called the longitudinals of the Block: and the others, its laterals.

V.

If, in a given Block, any rows, and as many columns, be selected: the square Block formed of their common Elements is called a Minor of the given Block⁸.

Hence any single Element of a Block, being common to one row and one column, is a Minor of it.

VI.

If n be that dimension of a Block which is not greater than the other: its Minors of the n^th degree are called its principal Minors; those of the $\overline{n-1}$^th degree its secondary Minors, and so on⁹.

Hence a square Block is its own principal Minor.

VII.

If, in a square Block, any rows, and as many columns, be selected: the Minor formed of their common Elements, and the Minor formed of the Elements common to the other rows and columns, are said to be complemental to each other¹⁰.

Conventions.

I.

Let it be agreed to represent the Elements of a square Block by symbols of the form $hk$, in which the first numeral indicates the row, and the second the column, to which the Element belongs. Thus, a Block of m rows and n columns may be represented thus:— $$\left\{\begin{array}{cccc}11,& 12& \cdots & 1n\\ 21,& 22& \cdots & 2n\\ ⋮& ⋮& & ⋮\\ m1,& m2& \cdots & mn\end{array}\right\}\text{.}$$

II.

And if it be required to represent 2 or more such Blocks, let them be distinguished by suffixing a certain letter to the symbol of each Block: e. g.:— $${\left\{\begin{array}{ccc}11& \cdots & 1n\\ ⋮& & ⋮\\ m1& \cdots & mn\end{array}\right\}}_a,{\left\{\begin{array}{ccc}11& \cdots & 1n\\ ⋮& & ⋮\\ m1& \cdots & mn\end{array}\right\}}_b\text{.}$$

III.

And if it be required to represent an Element of such a Block by itself, let it be distinguished by the same suffix: e. g., ${hk}_a$ represents an Element of the Block $${\left\{\begin{array}{ccc}11& \cdots & 1n\\ ⋮& & ⋮\\ m1& \cdots & mn\end{array}\right\}}_a\text{;}$$ ${hk}_b$ represents the corresponding Element of the Block $${\left\{\begin{array}{ccc}11& \cdots & 1n\\ ⋮& & ⋮\\ m1& \cdots & mn\end{array}\right\}}_b\text{.}$$

Definitions (continued)

VIII.

If there be a square Block of the n^th degree, and if all possible products be made of its Elements, taken n together, so that no product contain 2 Elements of the same row or of the same column; and if, representing the Elements of the Block by the symbols $$\left\{\begin{array}{ccc}11& \cdots & 1n\\ ⋮& & ⋮\\ m1& \cdots & mn\end{array}\right\}\text{,}$$ each such product be affected with + or −, according as the set of pairs of numerals, corresponding to that product, be of the even or the uneven class: the sum of these products, thus affected, is called the Determinant of the Block. And each of these products is called a Constituent of the Determinant.

IX.

The Constituent represented by the product $11.22\dots nn$ is called the Diagonal of the Determinant¹¹.

X.

If a square Block be such that its Determinant vanishes, or if an oblong Block be such that the Determinant of every one of its principal Minors vanishes: in either case the Block is said to be evanescent.

XI.

If there be a square Block, and if one of its Elements be selected; and if all the Constituents of its Determinant, which contain that Element, be collected together and formed into 2 factors, whereof that Element is one: the other factor is called the determinantal coefficient of that Element.

XII.

If 2 square Blocks be such that each Element of the second is equal to the determinantal coefficient of the corresponding Element of the first: the second Block is said to be adjugate to the first.

Conventions (continued)

IV.

Let it be agreed that the Determinant of a Block shall be represented by placing a perpendicular line on each side of it. Thus the Determinant of the Block $$\left\{\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& k\end{array}\right\}$$ will be represented by the symbol $$\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& k\end{array}\right|\text{.}$$

V.

If a Block be represented by the symbol $${\left\{\begin{array}{ccc}11& \cdots & 1n\\ ⋮& & ⋮\\ n1& \cdots & nn\end{array}\right\}}_a\text{,}$$ so that any Element of it is represented by a symbol of the form ${hk}_a$: let it be agreed that the determinantal coefficient of that Element shall be represented by a symbol of the form ${hk}_A$.

VI.

If there be 2 equally numerous sets of terms; and if each term of the one set be multiphed by the corresponding term of the other, and the products added: let it be agreed that this operation shall be denoted by placing the sign § between the symbols denoting the 2 sets.

Thus $(a_1,a_2,\dots a_n)\S (b_1,b_2,\dots b_n)=a_1{b}_1+a_2{b}_2+\dots +a_n{b}_n$.

Axioms.

I.

Each Constituent of the Determinant $$\left|\begin{array}{ccc}11& \cdots & 1n\\ ⋮& & ⋮\\ n1& \cdots & nn\end{array}\right|$$ contains n pairs of numerals, such that the antecedents are a certain permutation of the numbers 1 to n, as also are the consequents.

II.

If any row, or column, of a square Block be selected: each Constituent of the Determinant contains one term of that row or column.

Thus the Determinant $${\left|\begin{array}{ccc}11& \cdots & 1n\\ ⋮& & ⋮\\ n1& \cdots & nn\end{array}\right|}_a$$

$={11}_a.{11}_A+{12}_a.{12}_A+\dots +{1n}_a.{1n}_A$;

or $={11}_a.{11}_A+{21}_a.{21}_A+\dots +{n1}_a.{n1}_A$;

or $={h1}_a.{h1}_A+{h2}_a.{h2}_A+\dots +{hn}_a.{hn}_A$;

or $={1k}_a.{1k}_A+{2k}_a.{2k}_A+\dots +{nk}_a.{nk}_A$.

Hence if, in a square Block, the Elements of any one row, or column, be multiplied by v; the Determinant of the new Block is equal to that of the first multiplied by v.

And if the Elements of any q rows, or q columns, or r rows and s columns, where $\overline{r+s}=q$, be multiplied by v; the Determinant of the new Block is equal to that of the first multiplied by $v^q$.

And if each Element of any row, or column, be the sum of m terms: the Determinant may be expressed as the sum of m Determinants. For one Determinant may be formed, such that its corresponding row, or column, consists of the first terms of these sums; another, such that its corresponding row, or column, consists of the second terms of these sums; and so on.

Proposition I. Th.

The determinantal coefficient of any Element of a square Block is the Determinant of its complemental Minor, affected with + or − according as the numerals which constitute its symbol are similar or dissimilar.

Let the Block be represented by the symbol:— $$\left\{\begin{array}{ccc}11& \cdots & 1n\\ ⋮& & ⋮\\ n1& \cdots & nn\end{array}\right\}\text{;}$$ and call the selected Element $hk$;

then it is evident that the determinantal coefficient of $hk$ is the aggregate of all possible products of the Elements of its complemental Minor, taken $\overline{n-1}$ together, so that no product contain 2 Elements of the same row, or of the same column;

i. e. it is the aggregate of the Constituents of the Determinant of its complemental Minor; (Def. VIII.

also the sign of each such product is + or −, according as the corresponding set of pairs of numerals, taken along with the symbol $hk$ itself, is of the even or of the uneven class; (Def. VIII.

but if the symbol $hk$ be erased, the class, to which the remaining set belongs, is the same as that of the original set, or different, according as h and k are similar, or dissimilar; (Chap. I. Prop. V.

∴ the sign of each such product follows the Determinant law, or reverses it, according as h and k are similar, or dissimilar;

∴ the determinantal coefficient of any Element $hk$ is the Determinant of its complemental Minor, affected with + or −, according as h and k are similar or dissimilar.

Q. E. D¹².

Corollaries to Prop. I.

1.

If, in a square Block, any row, or column, be selected: the Determinant of the Block may be resolved into terms, each consisting of one of the Elements of that row, or column, multiplied by the Determinant of its complemental Minor¹³.

2.

If, in a square Block, the Elements in any one row, or column, all vanish but one: the Determinant of the Block is the product produced by multiplying the Determinant of the complemental Minor of that Element by that Element itself, affected with + or −, according as the numerals in its symbol are similar or dissimilar.

3.

Hence, if that Element be unity, the Determinant of the Block is the Determinant of the complemental Minor of that Element, affected with + or −, as before.

Proposition II. Th.

If, in a square Block, 2 rows, or 2 columns, be interchanged: the Determinant of the new Block has the same absolute value as that of the first, but the opposite sign.

Let the Block be represented by $${\left\{\begin{array}{ccc}11& \cdots & 1n\\ ⋮& & ⋮\\ n1& \cdots & nn\end{array}\right\}}_a\text{,}$$ and first let 2 rows be interchanged; call them the h^th and k^th rows; and let the new Block be represented by $${\left\{\begin{array}{ccc}11& \cdots & 1n\\ ⋮& & ⋮\\ n1& \cdots & nn\end{array}\right\}}_b\text{;}$$

$\therefore {hr}_a={kr}_b$, and ${kr}_a={hr}_b$, r taking any value $1\dots n$;

next let any Constituent of the Determinant of the original Block be selected; call it ${1a}_a\dots {hs}_a\dots {kt}_a\dots$; and let it $=M$;

now ${1a}_a={1a}_b$ and so for the other factors of it, with the exception of ${hs}_a$ and ${kt}_a$ which respectively $={ks}_b$ and ${ht}_b$;

$\therefore M={1a}_b\dots {ht}_b\dots {ks}_b\dots$, which is a Constituent of the new Block;

∴ for any Constituent M in the Determinant of the original Block, there is a Constituent in that of the new Block, of the same absolute value;

and the symbol representing the one may be deduced from that representing the other by one interchange of consequents;

∴ the 2 symbols are of different classes; (Ch. I. Prop. IV.

∴ the Constituents, represented by them, have opposite signs;

∴ the whole Determinants are equal in value, but have opposite signs.

Similarly, if 2 columns be interchanged.

Therefore, if there be &c., Q. E. D¹⁴.

Corollary to Prop. II.

If, in a square Block, a row, or a column, be made to pass over the next r rows, or columns, either way: the Determinant of the new Block has the same sign as that of the first, or the opposite sign, according as r is even or odd: that is, it is equal to the Determinant of the first Block multiplied by $(-1{)}^{r-1}$.

For this may be effected by interchanging it with each of these r rows, or columns, in turn; and after one such interchange, the sign of the Determinant is changed, after two, it is the same again, and so on¹⁵.

Proposition III. Th.

If, in a square Block, 2 rows, or 2 columns, be identical: the Determinant vanishes.

Call the Determinant “D”.

Now if the 2 identical rows, or columns, be interchanged, the Determinant of the new Block $=-D$; (Prop. II.

but the new Block is identical with the first;

$\therefore D=-D$;

i. e. $D=0$.

Therefore, if in a square Block, &c. Q. E. D¹⁶.

Corollaries to Prop. III.

1.

If a square Block of the n^th degree be represented by $${\left\{\begin{array}{ccc}11& \cdots & 1n\\ ⋮& & ⋮\\ n1& \cdots & nn\end{array}\right\}}_a\text{,}$$ and its adjugate Block by $${\left\{\begin{array}{ccc}11& \cdots & 1n\\ ⋮& & ⋮\\ n1& \cdots & nn\end{array}\right\}}_A\text{,}$$

then ${r1}_a.{s1}_A+{r2}_a.{s2}_A+\dots +{rn}_a.{sn}_A=0$,

and ${1r}_a.{1s}_A+{2r}_a.{2s}_A+\dots +{nr}_a.{ns}_A=0$,

so long as $r\ne s$. For the quantity ${s1}_a.{s1}_A+\text{\&c.}$ is the Determinant of the given Block, and if, for the several terms of the s^th row, there be substituted terms equal to those in the r^th row, this may be written ${r1}_a.{s1}_A+\text{\&c.}$; and since the new Block, so formed, has 2 rows identical, its Determinant vanishes.

2.

If, in a Block whose length exceeds its breadth by unity, the Elements of any one longitudinal be each multiplied by the Determinant of the Minor formed by erasing the lateral containing that Element: the sum of these products, affected with + and − alternately, is zero¹⁷.

3.

If, in a square Block, there be added to the several Elements of any row, or column, the corresponding Elements of any other row, or column, multiplied by any number: the Determinant of the new Block is the same as that of the first¹⁸.

Proposition IV. Th.

If there be a square Block, and if, retaining the first term of the first row in its place, the rows be made columns, and the columns rows: the Determinant of the new Block is equal to that of the first.

Let the Block be represented by $${\left\{\begin{array}{ccc}11& \dots & 1n\\ ⋮& & ⋮\\ n1& \dots & nn\end{array}\right\}}_a$$ and the new Block by $${\left\{\begin{array}{ccc}11& \dots & 1n\\ ⋮& & ⋮\\ n1& \dots & nn\end{array}\right\}}_b\text{;}$$ and let a certain Constituent of the original Block, arranged in order of antecedents, be represented by ${1\alpha }_a.{2\beta }_a\dots {n\zeta }_a$, where α, β, … ζ, are a certain permutation of the numbers 1 to n;

now ${1\alpha }_a={\alpha 1}_b$, ${2\beta }_a={\beta 2}_b$, … ${n\zeta }_a={\zeta n}_b$;

hence this Constituent $={\alpha 1}_b.{\beta 2}_b\dots {\zeta n}_b$;

that is, it = a certain Constituent of the new Block, arranged in order of consequents;

and the antecedents in the second case are the same permutation of the numbers 1 to n as the consequents in the first case;

∴ the two Constituents are of the same class; (Chap. I. Def. III.

∴ they have the same sign; (Chap. II. Def. VIII.

∴ for every Constituent of the original Block, there is one of the new Block, equal to it, and with the same sign;

∴ the two Determinants are equal.

Therefore, if there be, &c. Q. E. D¹⁹.

Proposition V. Th.

If there be given 2 Blocks, each consisting of n rows and r columns; and if each row of one Block be combined with each row of the other, by the process of multiplyhig the first term of one by the first term of the other, the second by the second, and so on, and adding the products; and if the $n^2$ quantities, so formed, be arranged as a square Block, in such a way that the Elements of the first row of the new Block are all formed from the first row of the first Block, by combining it with the n rows of the second Block successively, and so on; then

if $r<n$: the Determinant of the new Block vanishes:

if $r=n$: the Determinant of the new Block is the product of the Determinants of the 2 given Blocks:

if $r>n$: the Determinant of the new Block is the sum of all possible products that can be made, by taking any n columns of one of the given Blocks, in the order in which they stand, and the corresponding n columns of the other Block, and multiplying together the Determinants of the 2 Blocks so formed.

Let the 2 Blocks be represented by $${\left\{\begin{array}{ccc}11& \dots & 1r\\ ⋮& & ⋮\\ n1& \dots & nr\end{array}\right\}}_a,\text{ and }{\left\{\begin{array}{ccc}11& \dots & 1r\\ ⋮& & ⋮\\ n1& \dots & nr\end{array}\right\}}_b\text{;}$$ and the new Block by $${\left\{\begin{array}{ccc}11& \dots & 1n\\ ⋮& & ⋮\\ n1& \dots & nn\end{array}\right\}}_c\text{,}$$ wherein any Element ${hk}_c={\left\{h1\dots hr\right\}}_a\S {\left\{k1\dots kr\right\}}_b$; $$=\sum \left\{{h\alpha }_a.{k\alpha }_b\right\}\text{,}$$ in which α takes all values from 1 to r;

now let a certain Constituent of the Determinant of the new Block be arranged in order of antecedents, and be represented by ${1Q}_c.{2R}_c\dots {uT}_c$, in which Q, R, … T, are a certain permutation of the numbers 1 to n;

then this Constituent $$=\sum \{{1\alpha }_a.{Q\alpha }_b\}.\sum \{{2\beta }_a.{R\beta }_b\}\dots \sum \{{n\delta }_a.{T\delta }_b\}\text{;}$$ in which each of the quantities α, β, … δ, takes all values from 1 to r;

$$\begin{array}{rl}\therefore \text{ it}& =\sum \{{1\alpha }_a.{Q\alpha }_b.{2\beta }_a.{R\beta }_b\dots {n\delta }_a.{T\delta }_b\};\\ & =\sum \{{1\alpha }_a.{2\beta }_a\dots {n\delta }_a.{Q\alpha }_b.{R\beta }_b\dots {T\delta }_b\}\text{;}\end{array}$$

also this Constituent is affected with + or −, according as the series Q, R, … T, contains an even or odd number of derangements;

∴ the Determinant of the new Block $$=\sum \{{1\alpha }_a.{2\beta }_a\dots {n\delta }_a.{Q\alpha }_b.{R\beta }_b\dots {T\delta }_b\}\text{,}$$ in which not only does each of the quantities α, β, … δ, take all values from 1 to r, but also the series Q, R, … T, takes the values of all possible permutations of the numbers 1 to n;

$$\therefore \text{ it}=\sum \left\{{1\alpha }_a.{2\beta }_a\dots {n\delta }_a.\sum \left({Q\alpha }_b.{R\beta }_b\dots {T\delta }_b\right)\right\}\text{;}$$ wherein, whatsoever values are assigned to α, β, … δ, in the outer bracket, the same are assigned to them in the inner bracket:

now the sum $\sum \left({Q\alpha }_b.{R\beta }_b\dots {T\delta }_b\right)$, each term of which is affected with + or −, according as the series Q, R, … T, contains an even or odd number of derangements, is the Determinant $${\left|\begin{array}{ccc}1\alpha & \dots & 1\delta \\ ⋮& & ⋮\\ n\alpha & \dots & n\delta \end{array}\right|}_b\text{;}$$ that is, it is the Determinant of the square Block formed by taking from the b-Block its α^th column, its β^th column, and so on, until n columns have been taken, it being immaterial whether these be all different, or one or more of them be repeated any number of times.

∴ the Determinant of the new Block $$=\sum \left\{{1\alpha }_a.{2\beta }_a\dots n\delta .{\left|\begin{array}{ccc}1\alpha & \dots & 1\delta \\ ⋮& & ⋮\\ n\alpha & \dots & n\delta \end{array}\right|}_b\right\}\text{.}$$

let r be $<n$;

then it is not possible to take from the b-Block n different columns;

∴ the Determinant $${\left|\begin{array}{ccc}1\alpha & \dots & 1\delta \\ ⋮& & ⋮\\ n\alpha & \dots & n\delta \end{array}\right|}_b$$ always contains 2 identical columns;

∴ it always vanishes; (Prop. III.

∴ the Determinant of the new Block vanishes.

let $r=n$;

then, if the series α, β, … δ, be a permutation of the numbers 1 to n, the Determinant $${\left|\begin{array}{ccc}1\alpha & \dots & 1\delta \\ ⋮& & ⋮\\ n\alpha & \dots & n\delta \end{array}\right|}_b={\left|\begin{array}{ccc}11& \dots & 1n\\ ⋮& & ⋮\\ nn& \dots & nn\end{array}\right|}_b\text{,}$$ affected with +, or −, according as the series α, β, … δ, contains an even or odd number of derangements; for either of these may be obtained from the other by interchanging columns; hence the 2 Determinants have the same absolute magnitudes, and have the same sign, or not, according as their diagonals have the same sign or not;

but if the series α, β, … δ, be not such a permutation, the Determinant $${\left|\begin{array}{ccc}1\alpha & \dots & 1\delta \\ ⋮& & ⋮\\ n\alpha & \dots & n\delta \end{array}\right|}_b$$ vanishes as in the first case;

∴ the Determinant of the new Block $$\sum \left\{\pm {1\alpha }_a.{2\beta }_a\dots {n\delta }_a.{\left|\begin{array}{ccc}11& \dots & 1n\\ ⋮& & ⋮\\ nn& \dots & nn\end{array}\right|}_b\right\}\text{;}$$ $$=\sum \left\{\pm {1\alpha }_a.{2\beta }_a\dots {n\delta }_a\right\}.{\left|\begin{array}{ccc}11& \dots & 1n\\ ⋮& & ⋮\\ nn& \dots & nn\end{array}\right|}_b\text{;}$$ $$={\left|\begin{array}{ccc}11& \dots & 1n\\ ⋮& & ⋮\\ nn& \dots & nn\end{array}\right|}_a\times {\left|\begin{array}{ccc}11& \dots & 1n\\ ⋮& & ⋮\\ nn& \dots & nn\end{array}\right|}_b\text{;}$$ that is, it is the product of the Determinants of the 2 given Blocks²⁰.

let r be $>n$;

and let A, B, … N, be a certain set of n different numbers, selected from the numbers 1 to r, and orderly arranged; and let α, β, … δ, each take any of the values A, B, … N;

then, if the series α, β, … δ, be a permutation of the numbers A, B, … N, the Determinant $${\left|\begin{array}{ccc}1\alpha & \dots & 1\delta \\ ⋮& & ⋮\\ n\alpha & \dots & n\delta \end{array}\right|}_b={\left|\begin{array}{ccc}1A& \dots & 1N\\ ⋮& & ⋮\\ nA& \dots & nN\end{array}\right|}_b\text{,}$$ affected with + or −, according as the series α, β, … δ, contains an even or odd number of derangements;

but if the series α, β, … δ, be not such a permutation, the Determinant $${\left|\begin{array}{ccc}1\alpha & \dots & 1\delta \\ ⋮& & ⋮\\ n\alpha & \dots & n\delta \end{array}\right|}_b$$ vanishes as in the first case;

∴ the Determinant of the new Block contains the quantity $$\sum \left\{\pm {1\alpha }_a.{2\beta }_a\dots {n\delta }_a.{\left|\begin{array}{ccc}1A& \dots & 1N\\ ⋮& & ⋮\\ nA& \dots & nN\end{array}\right|}_b\right\}\text{;}$$

∴ it contains $\sum \left\{\pm {1\alpha }_a.{2\beta }_a\dots {n\delta }_a\right\}\times {\left|\begin{array}{ccc}1A& \dots & 1N\\ ⋮& & ⋮\\ nA& \dots & nN\end{array}\right|}_b$;

∴ it contains ${\left|\begin{array}{ccc}1A& \dots & 1N\\ ⋮& & ⋮\\ nA& \dots & nN\end{array}\right|}_a\times {\left|\begin{array}{ccc}1A& \dots & 1N\\ ⋮& & ⋮\\ nA& \dots & nN\end{array}\right|}_b$;

and the same thing may be proved for any other set of n different numbers, selected from the numbers 1 to r, and orderly arranged;

but if the series A, B, … N, though selected from the numbers 1 to r, be not all different numbers, the Determinant $$\left|\begin{array}{ccc}1A& \dots & 1N\\ ⋮& & ⋮\\ nA& \dots & nN\end{array}\right|$$ vanishes as in the first case;

∴ the Determinant of the new Block $$=\sum \left\{{\left|\begin{array}{ccc}1A& \dots & 1N\\ ⋮& & ⋮\\ nA& \dots & nN\end{array}\right|}_a\times {\left|\begin{array}{ccc}1A& \dots & 1N\\ ⋮& & ⋮\\ nA& \dots & nN\end{array}\right|}_b\right\}\text{,}$$ in which the series A, … N, take the values of every possible set of n different numbers, selected from the numbers 1 to r, and orderly arranged;

that is, it is the sum of all possible products that can be made, by taking any n columns of one of the given Blocks, in the order in which they stand, and the corresponding n columns of the other Block, and multiplying together the Determinants of the 2 Blocks so formed.

Therefore, if there be given 2 Blocks, &c. Q. E. D.

Corollary to Prop. V.

If $r=n$: then, in each of the given Blocks, rows may be made columns, and columns rows, without altering the Determinants; (Prop. IV.

∴ the new Block may be such that any Element of it, ${hk}_c$, has any one of the 4 values,

$$\left\{h1\dots hn\right\}\S {\left\{k1\dots kn\right\}}_b\text{,}$$

$$\left\{h1\dots hn\right\}\S {\left\{1k\dots nk\right\}}_b\text{,}$$

$$\left\{1h\dots nh\right\}\S {\left\{k1\dots kn\right\}}_b\text{,}$$

$$\left\{1h\dots nh\right\}\S {\left\{1k\dots nk\right\}}_b\text{.}$$

Proposition VI. Th.

If there be a square Block of the n^th degree: the Determinant of the adjugate Block is equal to the $\overline{n-1}$^th power of the Determinant of the first Block.

Let the Block be represented by $${\left\{\begin{array}{ccc}11& \dots & 1n\\ ⋮& & ⋮\\ n1& \dots & nn\end{array}\right\}}_a\text{,}$$ and the adjugate Block by $${\left\{\begin{array}{ccc}11& \dots & 1n\\ ⋮& & ⋮\\ n1& \dots & nn\end{array}\right\}}_A\text{;}$$ and let a Block of the n^th degree be formed, represented by $${\left\{\begin{array}{ccc}11& \dots & 1n\\ ⋮& & ⋮\\ n1& \dots & nn\end{array}\right\}}_c\text{,}$$ and such that any term ${hk}_c=({h1}_a\dots {hn}_a)\S ({k1}_A\dots {kn}_A)$;

and let their Determinants be represented by $D_a$, $D_A$, $D_c$.

Then $D_c=D_a.D_A$; (Prop. V.

now ${hk}_c={h1}_a.{k1}_A+\dots +{hn}_a.{kn}_A$;

∴ when $h=k$, ${hk}_c=D_a$; (Ax. II.

and when $h\ne k$, ${hk}_c=0$; (Prop. III. Cor. 1.

∴ all the Elements of $D_c$ vanish, except ${11}_c$, … ${nn}_c$, each of which $=D_a$;

∴ $D_c=D_a^n$;

i. e. $D_a.D_A=D_a^n$;

∴ $D_A=D_a^{n-1}$. Q. E. D²¹.

Proposition VII. Th.

If there be a square Block of the n^th degree, and if in it any Minor of the m^th degree be selected: the Determinant of the corresponding Minor in the adjugate Block is equal, in absolute magnitude, to the product of the $\overline{m-1}$^th power of the Determinant of the first Block, multiplied by the Determinant of the Minor complemental to the one selected.

Also, if the numerals, indicating the selected rows, be represented by α, β, …, and those indicating the selected columns by κ, λ, …; and their respective sums by $\sum (\alpha )$, $\sum (\kappa )$: the relationship of sign between the equal magnitudes will be secured by multiplying either of them by $(-1{)}^{m.(\sum \alpha +\sum \kappa )}$.

Let the Block be re-arranged, if necessary, by transposing rows and columns, so that the selected rows and columns shall stand first; and let it, when so arranged, be represented by $${\left\{\begin{array}{ccc}11& \dots & 1n\\ ⋮& & ⋮\\ n1& \dots & nn\end{array}\right\}}_a\text{,}$$ and its adjugate Block by $${\left\{\begin{array}{ccc}11& \dots & 1n\\ ⋮& & ⋮\\ n1& \dots & nn\end{array}\right\}}_A\text{;}$$ also let a Block of the n^th degree be formed, represented by $${\left\{\begin{array}{ccc}11& \dots & 1n\\ ⋮& & ⋮\\ n1& \dots & nn\end{array}\right\}}_b\text{,}$$ and such that its first m rows are identical with those of the A-Block, the rest of its diagonal consists of units, and all its other Elements are zero;

hence, ${\left\{\begin{array}{ccc}11& \dots & 1n\\ ⋮& & ⋮\\ n1& \dots & nn\end{array}\right\}}_b$ is identical with $$\left\{\begin{array}{cccccc}{11}_A& \dots & {1m}_A,& {1m+1}_A,& \dots & {1n}_A\\ ⋮& & ⋮& ⋮& & ⋮\\ {m1}_A& \dots & {mm}_A,& {mm+1}_A,& \dots & {mn}_A\\ 0& \dots & 0,& 1,& 0\dots & 0\\ ⋮& & \ddots & \ddots & \ddots & ⋮\\ ⋮& & & \ddots & \ddots & 0\\ 0& \dots & \dots & \dots & \dots 0,& 1\end{array}\right\}\text{;}$$

∴ $D_b=\left|\begin{array}{ccc}{11}_A& \dots & {1m}_A\\ ⋮& & ⋮\\ {m1}_A& \dots & {mm}_A\end{array}\right|$; (Prop. I. Cor. 3.

also let a Block of the n^th degree be formed, represented by $${\left\{\begin{array}{ccc}11& \dots & 1n\\ ⋮& & ⋮\\ n1& \dots & nn\end{array}\right\}}_c\text{,}$$ and such that any term ${hk}_c=({h1}_a\dots {hn}_a)\S ({k1}_b\dots {kn}_b)$;

∴ $D_a.D_b=D_c$, (Prop. V.

Now, for all values of $k\ngtr m$,

if $h=k$, ${hk}_c=D_a$; (Ax. II.

if $h\ne k$, ${hk}_c=0$; (Prop. III. Cor. 1.

i. e. in the first m columns of the c-Bloek, all the Elements vanish, except ${11}_c$, ${22}_c$, … ${mm}_c$, each of which $=D_a$;

also, for all values of $k>m$, $${hk}_c=({h1}_a,\dots {hk}_a,\dots {hn}_a)\S (0,\dots 1,\dots 0)\text{,}$$ since all the terms of the second series vanish, except ${kk}_b$, which $=1$;

∴ for all values of $k>m$, ${hk}_c={hk}_a$;

i. e. in the $\overline{m+1}$^th and following columns of the c-Block, the Elements are identical with the corresponding Elements of the a-Block;

i. e. ${\left\{\begin{array}{ccc}11& \dots & 1n\\ ⋮& & ⋮\\ n1& \dots & nn\end{array}\right\}}_c$ is identical with

$$\left\{\begin{array}{cccccc}{D}_a,0& \dots & 0,& {1m+1}_a& \dots & {1n}_a\\ 0& \ddots & ⋮& ⋮& & ⋮\\ ⋮& \ddots & 0& ⋮& & ⋮\\ ⋮& \ddots & D_a,& {mm+1}_a& \dots & {mn}_a\\ ⋮& & 0,& {m+1m+1}_a& \dots & {m+1n}_a\\ ⋮& & ⋮& ⋮& & ⋮\\ 0& \dots & 0,& {nm+1}_a& \dots & {nn}_a\end{array}\right\}\text{.}$$

∴ $D_c=D_a^m.\left|\begin{array}{ccc}{m+1m+1}_a& \dots & {m+1n}_a\\ ⋮& & ⋮\\ {nm+1}_a& \dots & {nn}_a\end{array}\right|$;

∴ $D_a.D_b=\text{do.}$

∴ $D_a.\left|\begin{array}{ccc}{11}_A& \dots & {1m}_A\\ ⋮& & ⋮\\ {m1}_A& \dots & {mm}_A\end{array}\right|=\text{do.}$

i. e. $\left|\begin{array}{ccc}{11}_A& \dots & {1m}_A\\ ⋮& & ⋮\\ {m1}_A& \dots & {mm}_A\end{array}\right|=D_a^{m-1}.\left|\begin{array}{ccc}{m+1m+1}_a& \dots & {m+1n}_a\\ ⋮& & ⋮\\ {nm+1}_a& \dots & {nn}_a\end{array}\right|$.

Now let the Determinant of the original Blocks before re-arrangement, be represented by ‘D’; the Determinant of the Minor, complemental to the one selected, by ‘C’; and if a Block be formed, adjugate to the original Block before re-arrangement, and if in it a Minor be taken corresponding to the one selected, let the Determinant of this Minor be represented by ‘J’.

Then, having regard to absolute magnitude only, $$D_a=D\text{;}$$

$$\left|\begin{array}{ccc}{m+1m+1}_a& \dots & {m+1n}_a\\ ⋮& & ⋮\\ {nm+1}_a& \dots & {nn}_a\end{array}\right|=C\text{;}$$

$$\left|\begin{array}{ccc}{11}_A& \dots & {1m}_A\\ ⋮& & ⋮\\ {m1}_A& \dots & {mm}_A\end{array}\right|=J\text{;}$$

$$J=\pm D^{m-1}.C\text{.}$$

Therefore, if there be, &c. Q. E. D²².

Secondly, the relationship of sign between these equal magnitudes will be secured by multiplying either by $(-1{)}^{m.(\sum (a)+\sum (\kappa ))}$.

Let the numeral, indicating the last of the selected rows, be represented by ‘ζ’.

Now, firstly, $D_a$ is equal to D in absolute magnitude; and it has the same, or a different, sign, according as the number of rows, over which other rows are transposed, together with the number of columns, over which other columns are transposed, is even or odd;

for each transposition over one row, or over one column, changes the sign of the Determinant; (Prop. II.

now the first of the selected rows, namely the α^th, is transposed over all the preceding $\overline{\alpha -1}$ rows;

the second, namely the β^th, is transposed over all the preceding $\overline{\beta -1}$ rows, excepting the α^th row itself; that is, it is transposed over $\overline{\beta -2}$ rows;

similarly the γ^th row is transposed over $\overline{\gamma -3}$ rows, and so on; and finally the ζ^th is transposed over $\overline{\zeta -m}$ rows;

∴ the number of rows, over which other rows are transposed, $$=\alpha +\beta +\dots +\zeta -(1+2+\dots +m)\text{;}$$ $$=\sum (\alpha )-\frac{m.(m+1)}{2}\text{;}$$

similarly, the number of columns, over which other columns are transposed, $$=\sum (\kappa )-\frac{m.(m+1)}{2}\text{;}$$

∴ their sum $=\sum (\alpha )+\sum (\kappa )-m.(m+1)$;

but $m.(m+1)$ is necessarily even;

∴ $D_a$ has the same sign as D, or a different one, according as $\sum (\alpha )+\sum (\kappa )$ is even or odd.

∴ $D_a=D.(-1{)}^{\sum (\alpha )+\sum (\kappa )}$;

∴ $D_a^{m-1}=D^{m-1}.(-1{)}^{(\sum (\alpha )+\sum (\kappa )).(m-1)}$.

Secondly, $\left|\begin{array}{ccc}{m+1m+1}_a& \dots & {m+1n}_a\\ ⋮& & ⋮\\ {nm+1}_a& \dots & {nn}_a\end{array}\right|$ is equal to C in absolute magnitude, and has the same sign;

for the two Determinants contain the same Elements, arranged in the same order.

Thirdly, $\left|\begin{array}{ccc}{11}_A& \dots & {1m}_A\\ ⋮& & ⋮\\ {m1}_A& \dots & {mm}_A\end{array}\right|$ is equal to J in absolute magnitude, and it has the same, or a different, sign, according as $(\sum (\alpha )+\sum (\kappa ))$ is even or odd;

for the Elements of the first of these Determinants are the determinantal coefficients of the Elements of the selected Minor, after the re-arrangement of the Block; and those of the other are the same coefficients before the re-arrangement;

hence the Elements of the one Determinant have the same absolute magnitude as those of the other, and the same, or a different, sign, according as the Determinant of the Block has, after the re-arrangement, the same, or a different, sign;

∴ $\left|\begin{array}{ccc}{11}_A& \dots & {1m}_A\\ ⋮& & ⋮\\ {m1}_A& \dots & {mm}_A\end{array}\right|=J.(-1{)}^{\sum (\alpha )+\sum (\kappa )}$.

Hence, substituting in the Equation already established, we get $$J.(-1{)}^{\sum (\alpha )+\sum (\kappa )}=D^{m-1}.C.(-1{)}^{(\sum (\alpha )+\sum (\kappa )).(m-1)}\text{;}$$

∴ $J=D^{m-1}.C.(-1{)}^{m.(\sum (\alpha )+\sum (\kappa ))}$²³.

Therefore, if the numerals, &c. Q. E. D.

Corollary to Prop. VII.

If $D_a=0$, the Determinant of any Minor in the adjugate Block $=0$, since it contains $D_a$, as a factor.

As a particular case of this, let $m=2$, and let the selected Minor contain the Elements common to the h^th and k^th rows and the r^th and s^th columns;

then $\left|\begin{array}{cc}{hr}_A,& {hs}_A\\ {kr}_A,& {ks}_A\end{array}\right|=0$;

i. e. ${hr}_A.{ks}_A={kr}_A.{hs}_A$;

∴ ${hr}_A:{hs}_A::{kr}_A.{ks}_A$;