5. The expansion of the array of the second order may be written out at a glance. The Process is less obvious, but still simple, for the array of the third order. It is as follows: Be neath the square array write the first and sec ond rows as shown in the figure. Then form the six products, each of three elements, traversed by one of the six oblique lines, apply ing the signs as indicated. The aggregate of terms thus obtained is the required expansion, as may readily be verified.
The reader will now do well to note how the values of the systems of unknowns .r, y, and .r, y, s,' obtained at the outset, may be written in the notation of determinants.
No such direct methods as the above are available for the expansion of determinant arrays of higher orders, but these will be con sidered further on. See 13.
• 6. In writing determinants it is often con venient to use a double-subscript notation, the first subscript designating the row and the sec ond the column to which the element belongs. Thus the element as stands in the third row and the fifth column. When the elements are merely symbolic it is customary to write only the principal terms between the vertical bars. In this, which is called the umbral notation, the determinant of the nth order is I ea." . . . (100 or I autta . . . ann I; which are often further abridged to I a,(n) I and I I respectively.
Thus far the economy of the notation of determinants is scarcely apparent. Specific forms of higher order have, however, been pur posely avoided. It is only necessary to write our the expansion of an array of the fourth order, which includes 41=24 terms each of the iourth degree, to understand the necessity of a general theory of such forms. Determinants of even the fifth and sixth orders would be, if written out in full, quite beyond manipulation; while the complete expansion of I ea,"as'ailyaaVa4VIC7VilasVinatilCaio'CanXICIPM I I and such functions are not at all uncommon, would fill over a thousand closely printed vol umes like the present! Yet, by means of the theory of determinants, such expressions are not only intelligible but manageable. The general properties of determinants will now be con sidered.
7. Any term of the development of I almil may be written ± • - - °An). (a) Designate by s the number of inversions in the permutation hi' . . . 1 and by v the number of Ineerchanges'of two elements necessary to bring the given term into the form ± ch(r)ch(e)ar) . . . (b)
Obviously s and v are either both even or both odd; but the permutation pqr . . . t is positive or negative, according as v is even or odd, and the term will, therefore, have the same sign whether it be determined by the permuta tion of the subscripts of (a) or by that of the superscripts of (6). It follows that the devel opment of a determinant may be obtained by permuting the superscripts and writing the signs of the terms in accordance with these permuta tions, instead of using the subscripts as already explained. Passing from one of these methods of development to the other is equivalent to changing each column of the array into a row of the same rank and vice versa. Hence, a determinant is not altered by changing the rows into corresponding columns and the columns into corresponding rows. Any statement made with reference to the rows of every determinant must, therefore, be equally true with respect to the columns. Rows and columns are alike called lines.
8. If any two parallel lines of a determinant be interchanged the determinant will be changed only in sign. For, interchanging two lines is the same as interchanging, in each term of the expansion, the indices corresponding to these lines. This reverses the sign of each term and therefore that of the whole determinant The element a may be transferred to the leading position by interchanging the rth row with the (k —1) preceding rows and the sth column with the (s-1) preceding columns. This being done, the resulting determinant must take the sign factor (-1) A determinant having two parallel lines iden tical is equal to zero; for the interchange of these identical lines reverses the sign without altering the value of the function.
9. A determinant having a line of elements each the sum of two or more quantities can be expressed as a sum of two or more deter minants. Let —= ba' — at(61— -±- . .)4...
(WI), ba• -± be such a determinant Then, writing Bi be--be . • any term of the development of A is of the form apbeer The terms d are obtained by permuting the subscripts p, q, r, . of apBer . . . . Permuting simultaneously the same subscripts in the second member and giving to each term thus obtained its appropriate sign, there results 160,4 ... 1=1016,4 ...