OF THE RECIPROCAL OF A DETERMINANT. 



629 



and Tf(n/) negative terms, all of which appear in the previous aggregate, — where in fact 



+ + 



the cofactor of x x x 2 . . . x n is | a 1 b 2 c 3 . . . j , — it follows that the sum or difference of the 



two is, like the former, an aggregate of n n positive terms, of which however |-(n/) are 



twins. If therefore this sum or difference is to be represented in the form 2/^ • D 1 , the 



number of terms in u x ■ D 1 must be n n ~ x and the number in D 1 must be n n ~ 2 . 



(23) The fact that the determinants of the type appearing in §§ 17, 20 have their 

 terms all positive and n n ~ 2 in number is a simple deduction from one of the following 

 pair of theorems : — 



(a) In the final expansion of the determinant 



<Z] + . . . + a z — a 1 — u., .... 



-ft & + ... + & -ft 



- 7i " y> 7i + • • • + 7* 



all the terms are positive, and the number of them is 



(z-n+l)(z+l)"-\ 

 (b) In the final expansion of the determinant 



ft & + ...+& -& 



y, - y, y l + . . • + y, 



all the terms are positive, and the number of them is 



(z+l)"- 1 . 

 If for a particular value of n the two theorems be established, it is easy to show 

 that the second will hold for the next higher value of n, and then from this to show 

 that the first will also hold for this higher value. In the one case the determinant is 

 expressed in terms of the elements of the first column and the complementary minors 

 of these elements : in the other, each element of the first column is increased by all the 

 elements in the same row with it, and the determinant then partitioned into determin- 

 ants with monomial elements in the first column. 



