DETEIIMINANT. — Imagine a square arrangement of terms, for :sample a", and taking this as the primitive arrangement, permute in every possible way entire columns (or, what would give the same result, entire lines) and for each such arrangement form the product of the terms in the dexter diagonal (Ice, to s.E) of the square, giving to such product the eign which belongs to the arrangement of the columns (or lines). The algebraical sum of these products is a determinant, and such determinant is, or may be represented as above, by inclosing the terms within two vertical lines. Thus the developed value of the determinant in question is The rule may be otherwise stated as follows : a determinant is the sum of a series of products each with its proper sign,such that in each product the factors are taken out of each line and out of each column, and if the factors are arranged according to tips primitive arrangement of the columns in which they occur, then the sign is that corresponding to the resulting arrangement of the lines (or rice versd); thus in the product — a e, the factors o,b",e occur in the columns 1, 2, 3 (they are therefore arranged according to the primitive arrangement of the columns) and in the lines 1, 3, 2. Such arrangement of the lines con sidered as derived from the primitive arrangement 1 2 3 is negative, and the product has therefore the sign—. A generalisation of this construction will bo mentioned under the term commutant.
The word rent/lane was formerly used as synonymous with determi nant, but it is now employed and is hero explained in a more extended signification. The new synonym climinant seems unnecessary.
A few of the numerous properties of determinants may be stated.
A determinant is a linear function (without constant term) of the terms in each of its columns, and also of the terms in each of its lines, or, more briefly expressed, it is a linear function of each column, and also of each line. Moreover, without altering the value of the deter minant, the lines may be made columns, and the columns lines, and all the properties of the function exist equally with respect to the lines and with respect to the columns. The absolute value of the determinant is not altered, but the sign is reversed, by an interchange of two columns, hence also if two columns become identical, the determinant vanishes. Moreover when the columns are permuted in any manner whatever, the absolute value is not altered, but the sign will be that corresponding to the arrangement of the columns. A
determinant may be developed as a linear function of the terms in any lino, thus— a, b I c' I +/Ple,a' I + c I I I a", ip", c" the sins being alternately positive and negative or else all positive, according as the number of columns is even or odd.
The square arrangement of terms out of which a determinant is formed, and generally any square or rectangular arrangement of terms, is called a matrix. Consider a determinant.
a, b, c, d dm and partitioning the lines in any manner, form with them the matrices a,1,, d n, b', d" , and out of those matrices, with complementary columns thereof, a sum of products 2-f- I I I I • (the sigii + being that corresponding to the product a of the terms in tEl dexter diagonals of the factor determinants, considered as a term of the original determinant), the sum of all the products so obtained is the original determinant.
It has been mentioned that the determinant is a linear function of each column ; hence if the terms of any column are pa, pa'.. the determinant le equal to p times a determinant in which the corres ponding column is a, a' ... and similarly if the column in a + b, a' + then the determinant in the sum of two other determinants in which the corresponding columns are a, a', ... and b, b; ... respectively.
This property, in combination with some of those already mentioned, leads very simply to the rule for the multiplication of determinants ; for example, we have o I I 13 I - 1Pa +43, p'a + p', a' a', Po' - pa' + + c/13' I from which the law is obvious. The product might also be'expressed, and although it appears less simple, there is an advantage in expressing it, in the form pa-l-era',03+(r0' I p'a+ tea', p' '13' I ' If we omit Simultaneously any line and any column of a determinant, and with the terms which are left forin a determinant, the determi nants so obtained are the first minors of the given determinant. A similar process, but omitting pairs, triads, &c. of lines and columns, gives the second minors ,third minors, &c. of,the given determinant. But the first minors are the most important, and are sometimes spoken of simply as the minors.