Page:EB1911 - Volume 01.djvu/661

From Wikisource
Jump to navigation Jump to search
This page has been proofread, but needs to be validated.
  
ALGEBRAIC FORMS
621

expression for the determinant becomes , viz. and are transposed, and it is clear that the number of transpositions necessary to convert the permutation of the second suffixes to the natural order is changed by unity. Hence the transposition of columns merely changes the sign of the determinant. Similarly it is shown that the transposition of any two columns or of any two rows merely changes the sign of the determinant.

Theorem.—Interchange of any two rows or of any two columns merely changes the sign of the determinant.

Corollary.—If any two rows or any two columns of a determinant be identical the value of the determinant is zero.

Minors of a Determinant.—From the value of we may separate those members which contain a particular element as a factor, and write the portion ; , the cofactor of , is called a minor of order of the determinant.

Now , wherein is not to be changed, but the second suffixes in the product assume all permutations, the number of transpositions necessary determining the sign to be affixed to the member.

Hence , where the cofactor of is clearly the determinant obtained by erasing the first row and the first column.

Hence

Similarly , the cofactor of , is shown to be the product of and the determinant obtained by erasing from the i th row and k th column. No member of a determinant can involve more than one element from the first row. Hence we have the development

,

proceeding according to the elements of the first row and the corresponding minors.

Similarly we have a development proceeding according to the elements contained in any row or in any column, viz.



This theory enables the evaluation of a determinant by successive reduction of the orders of the determinants involved.

Ex. gr.

Since the determinant

, having two identical rows,

vanishes identically; we have by development according to the elements of the first row

;

and, in general, since

,

if we suppose the ith and kth rows identical

;

and proceeding by columns instead of rows,

identical relations always satisfied by these minors.

If in the first relation of we write we find that so that breaks up into a sum of determinants, and we also obtain a theorem for the addition of determinants which have rows in common. If we multiply the elements of the second row by an arbitrary magnitude , and add to the corresponding elements of the first row, becomes , showing that the value of the determinant is unchanged. In general we can prove in the same way the—

Theorem.—The value of a determinant is unchanged if we add to the elements of any row or column the corresponding elements of the other rows or other columns respectively each multiplied by an arbitrary magnitude, such magnitude remaining constant in respect of the elements in a particular row or a particular column.

Observation.—Every factor common to all the elements of a row or of a column is obviously a factor of the determinant, and may be taken outside the determinant brackets.

Ex. gr.

The minor is , and is itself a determinant of order . We may therefore differentiate again in regard to any element where , ; we will thus obtain a minor of , which is a minor also of of order . It will be and will be obtained by erasing from the determinant the row and column containing the element ; this was originally the r th row and the sth column of ; the r th row of is the r th or (r–1)th row of according as and the sth column of is the sth or (s−1)th column of according as . Hence, if denote the number of transpositions necessary to bring the succession into ascending order of magnitude, the sign to be attached to the determinant arrived at by erasing the ith and r th rows and the k th and s th columns from in order produce will be raised to the power of .

Similarly proceeding to the minors of order , we find that is obtained from by erasing the i th, r th, t th, rows, the k th, s th, u th columns, and multiplying the resulting determinant by raised to the power and the general law is clear.

Corresponding Minors.—In obtaining the minor in the form of a determinant we erased certain rows and columns, and we would have erased in an exactly similar manner had we been forming the determinant associated with , since the deleting lines intersect in two pairs of points. In the latter case the sign is determined by raised to the same power as before, with the exception that , replaces ; but if one of these numbers be even the other must be uneven; hence

.

Moreover

,

where the determinant factor is given by the four points in which the deleting lines intersect. This determinant and that associated with are termed corresponding determinants. Similarly lines of deletion intersecting in points yield corresponding determinants of orders and respectively. Recalling the formula

,

it will be seen that and involve corresponding determinants. Since is a determinant we similarly obtain

,

and thence

;

and as before

,

an important expansion of .

Similarly

,

and the general theorem is manifest, and yields a development in a sum of products of corresponding determinants. If the jth column be identical with the ith the determinant vanishes identically; hence if be not equal to , or ,

.

Similarly, by putting one or more of the deleted rows or columns equal to rows or columns which are not deleted, we obtain, with Laplace, a number of identities between products of determinants of complementary orders.

Multiplication.—From the theorem given above for the expansion of a determinant as a sum of products of pairs of corresponding determinants it will be plain that the product of and may be written as a determinant of order , viz.


Multiply the 1st, 2nd ... nth rows by respectively, and