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9.2 DETERMINANTS OF ORDER 3

When the elimination technique is applied to the general system

of first-degree equations in the three unknowns x, y, and z, one obtains equations of the form

(2) Dx = E, Dy = F, Dz = G

where D is the three-by-three determinant (or determinant of order 3) defined by

In the equations above, E, F, and G are obtained by substituting the column of d's for the column of a's, b's, or c's, respectively, in D. For example,

If it follows from the equations (2) above that the system of simultaneous equations (1) has the unique solution

As in the case of 2 by 2 determinants, it can be shown that if D = 0, then the simultaneous equations (1) either have no common solution or an infinite number of common solutions. The determinant D is called the determinant of the system. The technique of expressing the solution of a system of simultaneous linear equations in terms of ratios of determinants when the determinant of the system is not zero, as in (3) above, is called Cramer's Rule.

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Wednesday, June 10, 1998