DETERMINANT, in mathematics, a function which presents itself in the solution of a system of simple equations.
1. Considering the equations
ax | + | by | + | cz | = | d, |
a′x | + | b′y | + | c′z | = | d′, |
a″x | + | b″y | + | c″z | = | d″, |
and proceeding to solve them by the so-called method of cross multiplication, we multiply the equations by factors selected in such a manner that upon adding the results the whole coefficient of y becomes = 0, and the whole coefficient of z becomes = 0; the factors in question are b′c″ − b″c′, b″c − bc″, bc′ − b′c (values which, as at once seen, have the desired property); we thus obtain an equation which contains on the left-hand side only a multiple of x, and on the right-hand side a constant term; the coefficient of x has the value
a(b′c″ − b″c′) + a′(b″c − bc″) + a″(bc′ − b′c),
and this function, represented in the form
a, | b, | c | , | ||
a′, | b′, | c′ | |||
a″, | b″, | c″ |
is said to be a determinant; or, the number of elements being 3^{2}, it is called a determinant of the third order. It is to be noticed that the resulting equation is
a, | b, | c | x = | d, | b, | c | ||||
a′, | b′, | c′ | d′, | b′, | c′ | |||||
a″, | b″, | c″ | d″, | b″, | c″ |
where the expression on the right-hand side is the like function with d, d′, d″ in place of a, a′, a″ respectively, and is of course also a determinant. Moreover, the functions b′c″ − b″c′, b″c − bc″, bc′ − b′c used in the process are themselves the determinants of the second order
b′, | c′ | , | b″, | c″ | , | b, | c | . | ||||||
b″, | c″ | b, | c | b′, | c′ |
We have herein the suggestion of the rule for the derivation of the determinants of the orders 1, 2, 3, 4, &c., each from the preceding one, viz. we have
a | = a, |
a, | b | = a | b′ | − a′ | b | . | ||||||
a′, | b′ |
a, | b, | c | = a | b′, | c′ | + a′ | b″, | c″ | + a″ | b, | c | , | ||||||||
a′, | b′, | c′ | b″, | c″ | b, | c | b′, | c′ | ||||||||||||
a″, | b″, | c″ |
a, | b, | c, | d | = a | b′, | c′, | d′ | − a′ | b″, | c″, | d″ | + a″ | b″′, | c″′, | d″′ | − a′″ | b, | c, | d | , | ||||||||||
a′, | b′, | c′, | d′ | b″, | c″, | d″ | b′″, | c′″, | d′″ | b, | c, | d | b′, | c′, | d′ | |||||||||||||||
a″, | b″, | c″, | d″ | b′″, | c′″, | d′″ | b, | c, | d; | b′, | c′, | d′ | b″, | c″, | d″ | |||||||||||||||
a′″, | b′″, | c′″, | d′″ |
and so on, the terms being all + for a determinant of an odd order, but alternately + and − for a determinant of an even order.
2. It is easy, by induction, to arrive at the general results:—
A determinant of the order n is the sum of the 1.2.3...n products which can be formed with n elements out of n ^{2} elements arranged in the form of a square, no two of the n elements being in the same line or in the same column, and each such product having the coefficient ± unity.
The products in question may be obtained by permuting in every possible manner the columns (or the lines) of the determinant, and then taking for the factors the n elements in the dexter diagonal. And we thence derive the rule for the signs, viz. considering the primitive arrangement of the columns as positive, then an arrangement obtained therefrom by a single interchange (inversion, or derangement) of two columns is regarded as negative; and so in general an arrangement is positive or negative according as it is derived from the primitive arrangement by an even or an odd number of interchanges. [This implies the theorem that a given arrangement can be derived from the primitive arrangement only by an odd number, or else only by an even number of interchanges,—a theorem the verification of which may be easily obtained from the theorem (in fact a particular case of the general one), an arrangement can be derived from itself only by an even number of interchanges.] And this being so, each product has the sign belonging to the corresponding arrangement of the columns; in particular, a determinant contains with the sign + the product of the elements in its dexter diagonal. It is to be observed that the rule gives as many positive as negative arrangements, the number of each being = 12 1.2...n.
The rule of signs may be expressed in a different form. Giving to the columns in the primitive arrangement the numbers 1, 2, 3 ... n, to obtain the sign belonging to any other arrangement we take, as often as a lower number succeeds a higher one, the sign −, and, compounding together all these minus signs, obtain the proper sign, + or − as the case may be.
Thus, for three columns, it appears by either rule that 123, 231, 312 are positive; 213, 321, 132 are negative; and the developed expression of the foregoing determinant of the third order is
= ab′c″ − ab″c′ + a′b″c − a′bc″ + a″bc′ − a″b′c.
3. It further appears that a determinant is a linear function^{[1]} of the elements of each column thereof, and also a linear function of the elements of each line thereof; moreover, that the determinant retains the same value, only its sign being altered, when any two columns are interchanged, or when any two lines are interchanged; more generally, when the columns are permuted in any manner, or when the lines are permuted in any manner, the determinant retains its original value, with the sign + or − according as the new arrangement (considered as derived from the primitive arrangement) is positive or negative according to the foregoing rule of signs. It at once follows that, if two columns are identical, or if two lines are identical, the value of the determinant is = 0. It may be added, that if the lines are converted into columns, and the columns into lines, in such a way as to leave the dexter diagonal unaltered, the value of the determinant is unaltered; the determinant is in this case said to be transposed.
4. By what precedes it appears that there exists a function of the n ^{2} elements, linear as regards the terms of each column (or say, for shortness, linear as to each column), and such that only the sign is altered when any two columns are interchanged; these properties completely determine the function, except as to a common factor which may multiply all the terms. If, to get rid of this arbitrary common factor, we assume that the product of the elements in the dexter diagonal has the coefficient +1, we have a complete definition of the determinant, and it is interesting to show how from these properties, assumed for the definition of the determinant, it at once appears that the determinant is a function serving for the solution of a system of linear equations. Observe that the properties show at once that if any column is = 0 (that is, if the elements in the column are each = 0), then the determinant is = 0; and further, that if any two columns are identical, then the determinant is = 0.
5. Reverting to the system of linear equations written down at the beginning of this article, consider the determinant
ax | + | by | + | cz | − | d, | b, | c | ; | ||
a′x | + | b′y | + | c′z | − | d′, | b′, | c′ | |||
a″x | + | b″y | + | c″z | − | d″, | b″, | c″ |
it appears that this is
= x | a, | b, | c | + y | b, | b, | c | + z | c, | b, | c | − | d, | b, | c | ; | ||||||||
a′, | b′, | c′ | b′, | b′, | c′ | c′, | b′, | c′ | d′, | b′, | c′ | |||||||||||||
a″, | b″, | c″ | b″, | b″, | c″ | c″, | b″, | c″ | d″, | b″, | c″ |
viz. the second and third terms each vanishing, it is
= x | a, | b, | c | − | d, | b, | c | . | ||||
a′, | b′, | c′ | d′, | b′, | c′ | |||||||
a″, | b″, | c″ | d″, | b″, | c″ |
But if the linear equations hold good, then the first column of the original determinant is＝0, and therefore the determinant itself is＝0; that is, the linear equations give
x | a, | b, | c | − | d, | b, | c | ＝0; | ||||
a′, | b′, | c′ | d′, | b′, | c′ | |||||||
a″, | b″, | c″ | d″, | b″, | c″ |
which is the result obtained above.
We might in a similar way find the values of y and z, but there is a more symmetrical process. Join to the original equations the new equation
αx + βy + γz ＝ δ;
a like process shows that, the equations being satisfied, we have
α, | β, | γ, | δ | ＝0; | ||
a, | b, | c, | d | |||
a′, | b′, | c′, | d′ | |||
a″, | b″, | c″, | d″ |
or, as this may be written,
α, | β, | γ, | − δ | a, | b, | c | ＝0; | |||||
a, | b, | c, | d | a′, | b′, | c′ | ||||||
a′, | b′, | c′, | d′ | a″, | b″, | c″ | ||||||
a″, | b″, | c″, | d″ |
which, considering δ as standing herein for its value αx + βy + γz, is a consequence of the original equations only: we have thus an expression for αx + βy + γz, an arbitrary linear function of the unknown quantities x, y, z; and by comparing the coefficients of α, β, γ on the two sides respectively, we have the values of x, y, z; in fact, these quantities, each multiplied by
a, | b, | c | , | ||
a′, | b′, | c′ | |||
a″, | b″, | c″ |
are in the first instance obtained in the forms
1 | , | 1 | , | 1 | ; | |||||||||||||||
a, | b, | c, | d | a, | b, | c, | d | a, | b, | c, | d | |||||||||
a′, | b′, | c′, | d′ | a′, | b′, | c′, | d′ | a′, | b′, | c′, | d′ | |||||||||
a″, | b″, | c″, | d″ | a″, | b″, | c″, | d″ | a″, | b″, | c″, | d″ |
but these are
＝ | b, | c, | d | , − | c, | d, | a | , | d, | a, | b | , | ||||||
b′, | c′, | d′ | c′, | d′, | a′ | d′, | a′, | b′ | ||||||||||
b″, | c″, | d″ | c″, | d″, | a″ | d″, | a″, | b″ |
or, what is the same thing,
＝ | b, | c, | d | , | c, | a, | d | , | a, | b, | d | ||||||
b′, | c′, | d′ | c′, | a′, | d′ | a′, | b′, | d′ | |||||||||
b″, | c″, | d″ | c″, | a″, | d″ | a″, | b″, | d″ |
respectively.
6. Multiplication of two Determinants of the same Order.—The theorem is obtained very easily from the last preceding definition of a determinant. It is most simply expressed thus—
(α, α′, α″), | (β, β′, β″), | (γ, γ′, γ″) | |||||||||||||||||||
(a, | b, | c | ) | ” | ” | ” | = | a, | b, | c | . | α, | β, | γ | , | ||||||
(a′, | b′, | c′ | ) | ” | ” | ” | a′, | b′, | c′ | α′, | β′, | γ′ | |||||||||
(a″, | b″, | c″ | ) | ” | ” | ” | a″, | b″, | c″ | α″, | β″, | γ″ |
where the expression on the left side stands for a determinant, the terms of the first line being (a, b, c)(α, α′, α″), that is, aα + bα′ + cα″, (a, b, c)(β, β′, β″), that is, aβ + bβ′ + cβ″, (a, b, c)(γ, γ′, γ″), that is aγ + bγ′ + cγ″; and similarly the terms in the second and third lines are the life functions with (a′, b′, c′) and (a″, b″, c″) respectively.
There is an apparently arbitrary transposition of lines and columns; the result would hold good if on the left-hand side we had written (α, β, γ), (α′, β′, γ′), (α″, β″, γ″), or what is the same thing, if on the right-hand side we had transposed the second determinant; and either of these changes would, it might be thought, increase the elegance of the form, but, for a reason which need not be explained,^{[2]} the form actually adopted is the preferable one.
To indicate the method of proof, observe that the determinant on the left-hand side, qua linear function of its columns, may be broken up into a sum of (3^{3}＝) 27 determinants, each of which is either of some such form as
＝αβγ′ | a, | a, | b | , | ||
a′, | a′, | b′ | ||||
a″, | a″, | b″ |
where the term αβγ′ is not a term of the αβγ-determinant, and its coefficient (as a determinant with two identical columns) vanishes; or else it is of a form such as
＝αβ′γ″ | a, | b, | c | , | ||
a′, | b′, | c′ | ||||
a″, | b″, | c″ |
that is, every term which does not vanish contains as a factor the abc-determinant last written down; the sum of all other factors ± αβ′γ″ is the αβγ-determinant of the formula; and the final result then is, that the determinant on the left-hand side is equal to the product on the right-hand side of the formula.
7. Decomposition of a Determinant into complementary Determinants.—Consider, for simplicity, a determinant of the fifth order, 5＝2 + 3, and let the top two lines be
a, | b, | c, | d, | e |
a′, | b′, | c′, | d′, | e′ |
then, if we consider how these elements enter into the determinant, it is at once seen that they enter only through the determinants of the second order
a, | b | , | ||
a′, | b′ |
&c., which can be formed by selecting any two columns at pleasure. Moreover, representing the remaining three lines by
a″, | b″, | c″, | d″, | e″ |
a‴, | b‴, | c‴, | d‴, | e‴ |
a⁗, | b⁗, | c⁗, | d⁗, | e⁗ |
it is further seen that the factor which multiplies the determinant formed with any two columns of the first set is the determinant of the third order formed with the complementary three columns of the second set; and it thus appears that the determinant of the fifth order is a sum of all the products of the form
a, | b | c″, | d″, | e″ | , | |||||
a′, | b″ | c‴, | d‴, | e‴ | ||||||
c⁗, | d⁗, | e⁗ |
the sign ± being in each case such that the sign of the term ± ab′c″d‴e⁗ obtained from the diagonal elements of the component determinants may be the actual sign of this term in the determinant of the fifth order; for the product written down the sign is obviously +.
Observe that for a determinant of the n-th order, taking the decomposition to be 1 + (n − 1), we fall back upon the equations given at the commencement, in order to show the genesis of a determinant.
8. Any determinant
a, | b | ||
a′, | b′ |
formed out of the elements of the original determinant, by selecting the lines and columns at pleasure, is termed a minor of the original determinant; and when the number of lines and columns, or order of the determinant, is n−1, then such determinant is called a first minor; the number of the first minors is＝n ^{2}, the first minors, in fact, corresponding to the several elements of the determinant—that is, the coefficient therein of any term whatever is the corresponding first minor. The first minors, each divided by the determinant itself, form a system of elements inverse to the elements of the determinant.
A determinant is symmetrical when every two elements symmetrically situated in regard to the dexter diagonal are equal to each other; if they are equal and opposite (that is, if the sum of the two elements be＝0), this relation not extending to the diagonal elements themselves, which remain arbitrary, then the determinant is skew; but if the relation does extend to the diagonal terms (that is, if these are each＝0), then the determinant is skew symmetrical; thus the determinants
a, | h, | g | ; | a, | ν, | −μ | ; | 0, | ν, | −μ | ||||||
h, | b, | f | −ν, | b, | λ | −ν, | 0, | λ | ||||||||
g, | f, | c | μ, | −λ, | c | μ, | −λ, | 0 |
are respectively symmetrical, skew and skew symmetrical.
The theory admits of very extensive algebraic developments, and applications in algebraical geometry and other parts of mathematics. For further developments of the theory of determinants see Algebraic Forms. (A. Ca.)
9. History.—These functions were originally known as “resultants,” a name applied to them by Pierre Simon Laplace, but now replaced by the title “determinants,” a name first applied to certain forms of them by Carl Friedrich Gauss. The germ of the theory of determinants is to be found in the writings of Gottfried Wilhelm Leibnitz (1693), who incidentally discovered certain properties when reducing the eliminant of a system of linear equations. Gabriel Cramer, in a note to his Analyse des lignes courbes algébriques (1750), gave the rule which establishes the sign of a product as plus or minus according as the number of displacements from the typical form has been even or odd. Determinants were also employed by Étienne Bezout in 1764, but the first connected account of these functions was published in 1772 by Charles Auguste Vandermonde. Laplace developed a theorem of Vandermonde for the expansion of a determinant, and in 1773 Joseph Louis Lagrange, in his memoir on Pyramids, used determinants of the third order, and proved that the square of a determinant was also a determinant. Although he obtained results now identified with determinants, Lagrange did not discuss these functions systematically. In 1801 Gauss published his Disquisitiones arithmeticae, which, although written in an obscure form, gave a new impetus to investigations on this and kindred subjects. To Gauss is due the establishment of the important theorem, that the product of two determinants both of the second and third orders is a determinant. The formulation of the general theory is due to Augustin Louis Cauchy, whose work was the forerunner of the brilliant discoveries made in the following decades by Hoëné-Wronski and J. Binet in France, Carl Gustav Jacobi in Germany, and James Joseph Sylvester and Arthur Cayley in England. Jacobi’s researches were published in Crelle’s Journal (1826–1841). In these papers the subject was recast and enriched by new and important theorems, through which the name of Jacobi is indissolubly associated with this branch of science. The far-reaching discoveries of Sylvester and Cayley rank as one of the most important developments of pure mathematics. Numerous new fields were opened up, and have been diligently explored by many mathematicians. Skew-determinants were studied by Cayley; axisymmetric-determinants by Jacobi, V. A. Lebesque, Sylvester and O. Hesse, and centro-symmetric determinants by W. R. F. Scott and G. Zehfuss. Continuants have been discussed by Sylvester; alternants by Cauchy, Jacobi, N. Trudi, H. Nagelbach and G. Garbieri; circulants by E. Catalan, W. Spottiswoode and J. W. L. Glaisher, and Wronskians by E. B. Christoffel and G. Frobenius. Determinants composed of binomial coefficients have been studied by V. von Zeipel; the expression of definite integrals as determinants by A. Tissot and A. Enneper, and the expression of continued fractions as determinants by Jacobi, V. Nachreiner, S. Günther and E. Fürstenau. (See T. Muir, Theory of Determinants, 1906).
- ↑ The expression, a linear function, is here used in its narrowest sense, a linear function without constant term; what is meant is that the determinant is in regard to the elements a, a′, a″, . . . of any column or line thereof, a function of the form Aa + A′a′ + A″a″ + . . . . without any term independent of a, a′, a″ . . .
- ↑ The reason is the connexion with the corresponding theorem for the multiplication of two matrices.