Modern Higher Algebra. 



BY E. MILLER. 



Note. — The Theory of Determinants, as a branch of Modern High- 

 er Algebra, has taken its appropriate place in all of the best works on 

 Higher Algebra. It is an instrument of great utility and power in 

 the most complex mathematical operations, and as illustrative of its 

 methods, the following problems, algebraic in their character, are 

 herewith demonstrated. They were originally suggested by a French 

 teacher of mathematics. 



I. 



Show that if a determinant equals zero, the minors corresponding to 

 tke elements of two parallel rows are proportional, and reciprocally. 



SOLUTION. 



Let (a{a| . , . . a^) be a determinant, D, whose value is zero. 



We shall consider a homogeneous system of 7i equations of n un- 

 known quantities, in which the coefficient of X\ in the /&th equation is 

 a^-, the solution of these equations yields values of the unknown 

 quantities other than zero. 



Suppose that one at least of the coefficients of the/th row and one 

 of the coefficients of the elements of the ^th row have other values 

 than zero. 



By making Bj to be the coefficient of the element a?, we know that 

 the solution is general; so that 



(I), x,=kB^, x,=kB|, . . . . , x^=kBn 



in which k may be any number. 



In a similar manner u being an arbitrary number, 

 (2), , Xj=uB^, x^^uBJ, , x^^uBn, 



with one value of k in (i) corresponding to a value for u in (2) such 

 that we have the following equations, 



(3), kBi=uBS kB3=:uB2, , kBn=uBn. 



^^'" P q P q P q 



What has thus been established with regard to the rows can, by a 

 similar process, be shown to be true of the columns of a determinant. 



Reciprocally, if the equations (3) hold between the elements of two 

 rows, then the determinant J, which is the adjugateof D, has two par- 



(133) KAN. UNIV. QUAR., VOL. I., NO. 3, JAN., 1893. 



