DETERMINANT, a special kind of algebraic expression in volving a square number of quantities or elements, and usually denoted by arranging these elements in a square array with an upright line on each side: A determinant with elements is said to be of the nth order, those just written being of the first, second and third orders re spectively. It consists of the aggregate of all possible products of n of these elements taken one from each horizontal row and vertical column of the square array, with the sign -I- or — attached according to the rule of signs given below. The par ticular product whose factors are the elements in the principal diagonal, extending from the upper left-hand to lower right-hand corner of the array, always has the sign + attached.
The number of terms in a determinant increases rapidly with its order. Thus a determinant of the fourth order has 24 terms, a determinant of the fifth order has I 20 terms, and so on.
If any k rows and k columns of the determinantal array be selected, the elements common to these rows and columns form an array of elements and define a minor determinant, or simply a minor, of order k. The minor formed from the n-k rows and n-k columns not selected is called the complementary minor. Rela tions between a determinant and its minors play an important role in the theory of determinants.
The discovery of determinants is usually ascribed to G. W. Leibniz, who stated their law of formation in a letter written to De L'Hospital in 1693 ; however, a Japanese mathematician Seki Kowa, had come close to the same discovery at least as early as 1683. The work of Leibniz seems to have exerted no influence.
It was left to G. Cramer to rediscover determinants, and to publish first a statement of their law of formation in 175o. The convenient notation used above was introduced by A. Cayley in 1841. For these and other historical facts the reader may be referred to T. Muir's The Theory of Determinants in the Histor ical Order of Development, vols. i.-iv. (London, 1906-20), and to D. E. Smith's and Y. Mikami's A History of Japanese Mathe matics (Chicago, 1914)• Determinants necessarily arose as soon as two or more equa tions of the first degree in the unknown quantities were con sidered from an algebraic point of view. For example any two equations of the first degree may be written ax+by=e, cx+d y = f , in which the letters a, b, ... f stand for known quantities, while x and y are the two unknown quantities to be determined. If we multiply both sides of the first of these equations by d, and multiply both sides of the second equation by b, we obtain adx+bdy=ed, bcx+bdy=bf, whence, by subtraction, there results (ad — bc) x = (ed — bf) or a d x = e f d • Hence x can be expressed as the quotient of two determinants of the second order, at least provided (ad-bc) is not zero.
Similarly if there be given any number n of equations of the first degree in an equal number of unknown quantities, x, y, z, ... , these quantities can be expressed as quotients of determi nants of the nth order. Such expressions for the quantities x, y, z ... , are obtained below.
Determinants are especially useful as an instrument of classi fication. Thus in dealing with equations of the first degree many possibilities exist, of which one, for instance, is that in which there are no values whatsoever of the unknown quantities satis fying the equations. The theory of determinants gives a com prehensive means of distinguishing between the various cases which may arise in this and many other algebraic questions.
The subject of determinants forms an extensive and funda mentally important part of higher algebra, and has application in nearly every mathematical field. The reader interested in the classical applications of determinants in higher algebra may be referred to M. Bocher's Introduction to Higher Algebra (New York, 1907).
The daring notion of a determinant of infinite order was first developed by the American mathematician and astronomer, G. W. Hill (1877), in connection with his theory of lunar motion. This extension has proved to be one of importance. On the other hand determinants of n dimensions, based upon cubical arrays of elements (n=3) or, more generally on arrays of n dimensions, do not seem to be especially useful.
In illustration of this rule of signs we may consider the product b h k m in the determinant of the fourth order The four pairs of letters bm, hk, hm, km, are to be counted: b appears to the right of and above on; h, to the right of and above k ; etc. Hence the rule attaches a sign + to this one of the 24 products which appear in A.
The Fundamental Property of Determinants.—Imagine two adjacent rows or columns of a determinant to be interchanged. The only pair of elements of any product whose status changes as far as the rule of signs is concerned, is precisely the pair found in the two rows or columns which are interchanged. If this pair was counted before the interchange it will not be counted afterwards, and if not counted bef ore, it will be counted afterwards. Hence such an interchange changes the number of pairs to be counted from odd to even, or from even to odd, and thus alters the sign of every product in the determinant.
Now to interchange any two rows or columns of a determinant requires an odd number of interchanges of adjacent pairs of rows or of columns. For example, in order to interchange the first and last rows in the above determinant A, the first row may be moved to the last position by three such interchanges of adjacent rows, and the row that stood last at the outset goes to the first position by two more such interchanges, so that five inter changes of adjacent rows suffice. In general, if two rows or columns are separated by k intervening rows or columns, evidently the odd number 2k-I of interchanges of adjacent rows or columns will suffice. But each interchange of adjacent rows or columns alters the sign of every term in the determinant.
Thus the determinant has the fundamental property that the interchange of any two rows or columns reverses its sign.
In consequence, if two rows or two columns are made up of the same elements in the same order, the determinant must reduce to zero. For by the interchange of these two rows or columns the determinant is unaltered and yet is changed to its negative, and o is the only number equal to its negative.
The Solution of Equations of the First Degree.—Let us now return to the determinant of the fourth order written above, which can be regarded as associated with the four equations of the first degree, in x, y, z, w, written below: ax+by+cz+dw = q, ex+f y-}-gz+hw=r, ix+jy+kz+lw=s, mx+ny+oz+pw= t, in which a, b, ..., t are regarded as known quantities, and x, y, z, w, are unknown quantities to be determined. We propose to show that if A stands for the determinant written above, and z 1, A3, L 4 stand for the f our similar determinants obtained by replacing the first, second, third and fourth columns of A respectively by the column formed from the known quantities q, r, s, t, then the solution is given by the formulas: x = Al/A, y = z = w = Let us denote the aggregate of all the products in A which contain a factor a by aA; similarly we may define bB, cC, ..., pP. Since every product in A contains one and only one element in the first column, we must have: aA-FeE-FiI+mM=O.
This gives a decomposition of A in four parts ; there are evi dently seven other like decompositions of A obtained by con sidering the elements of any other column or of any row.
Now if we replace a, e, m, in the above equation by the elements of any other column of A, say by b, f, j, it respectively, the sum vanishes, i.e., bA+ f E+jl+nM In fact this sum is evidently itself a determinant, obtained from A by replacing et, e, i, m, by b, f, j, n, respectively, and so its first two columns are the same. Consequently this sum will vanish.
Now multiply through the first of the four given equations by A, the second by E, the third by I, the fourth by M, and add all four equations. The co-efficient of x in the resulting equation is A. The co-efficient of y is the sum written above which vanishes. The coefficients of z and w are similar sums which vanish also. On the right-hand side there is a sum which is evidently the determinant A,. Thus we find Ox = from which follows the expression for x as stated.
To obtain the expression for y it suffices to multiply the equa tions by B, F, J, N, respectively and add. A similar method expresses z and w in this manner.
Evidently this reasoning would establish the following general rule: Let there be given any set of equations of the first degree with terms arranged according to the n unknown quantities in the left-hand members of the equations, while the terms on the right are known. If the determinant A formed from the square array of coefficients on the left-hand side is not o, and if A2, ..., A denote the n determinants obtained by replacing the first, second, ... , nth columns of A respectively by the sets of n constants on right-hand sides of the equations, then the unknown quantities are given by the quotients of determinants . , This general rule applies in case A is not zero, and actually furnishes the only possible set of values of the unknown quan tities. No further consideration of the exceptional case when A is zero can be included here. (G. D. Br.)