THE PRODUCT OF TWO DETERMINANTS OF THE THIRD ORDER

755. The rule for forming the product of two determinants of any order will be found by verifying the following identity which expresses the product of two determinants of the third order:

This identity may be verified by means of the addition theorem, 8753. If the last determinant is developed by the addition theorem, we obtain 27 determinants of the following types:

Of these determinants, (1) is zero, since on removing the factors b1, c, the columns are identical; similarly, on removing the factors 41, b 2, c2 two columns of the determinant (2) are the same and therefore the determinant (2) is 0; determinants (3) and (5) may be written

2

2

while determinant (4) is zero for the same reason that (3) is zero. If, now, determinant (a) was developed by the usual method, determinant (3) would be the product of the determinant (1) times the element of the leading diagonal a, b, c, and (5) would be the product of (7) by the second negative term, b, c, a,, of the development of (a), etc.

2

RULE. To form the product T, of two determinants T, and T2, first connect with plus signs the constituents in the rows of both the determinants T and T. Then place the first row of T upon each row of T in order and let each pair of constituents as they touch become products. This is the first column of T.

2

2

Perform the same operation upon T2 with the second row of T obtain the second column of T, and so on.

3

to

(1)

DETERMINANTS OF THE FOURTH ORDER

756. Consider the system of four linear homogeneous equations

a1x + b1y + c1z + d ̧v = 0

ax+by+cz+d ̧v=0

azc + by + cqz dr=0

x + b1y + c1 z + d ̧v = 0.

аx

Solving the last three equations for x, y, and z by 8752 we have

The eliminant of the given system is found by substituting these values of x, y, and z in the first equation; it is

The first member of this equation is called the determinant of the fourth order and is written

The coefficients of a, b, c, d, in equation (3) taken with their proper signs are respectively the minors of a, b, c, d, found by striking out the row and the column which respectively have their constituents common, and are represented by A1, B1, C1, D1.

757. Since it is not the purpose of this text to study determinants of an order higher than the fourth we call attention to the fact that the properties of determinants which have been established for those of the second and the third orders are general and may be extended to determinants of any order.

758. The determinant of the fourth order in (4) can be developed by means of equation (3); the development will be found to be

We illustrate the usual method for reducing a determinant of the

Subtract the fourth column from the second and the third and obtain

759. We have already learned how to find the eliminant of a system of linear equations of two, three, and four unknown quantities connected respectively by two, three, and four equations. 760. Another important problem in elimination can be solved by the use of determinants, namely:

By means of Sylvester's Dialytic Method we can find the rational eliminant for the case when two equations are of any degree.

EXAMPLE 1. Find the eliminant of the two equations of the second degree,

(1) ax2 + bx + c = 0 and (2) a'x2+ b'x + c' = 0. Multiply both (1) and (2) by x2, x; then we have the following system of equations:

We may regard these equations as linear and homogeneous in the four variables x1, x3, x2, x considered separately. Hence their eliminant is

Multiply (1) successively by 2, x, and (2) by x3, x2, x. have the following systems of equations and their eliminant:

Then we