30 



ALM AR NÆSS. 



M.-N. Kl. 



'Ih.n 



(15) 



0/» 



Summing for all possible sets ol" I//). As j and / are: not quite inde- 

 pendent of one another \j " I], it is convenient to consider _/' / in this and 



... . .... • r . /"I " ^" ^ 1 ' 



similar summations as a single index running from 1 to 



The '^-^^ (|uantities: d^i ^ — (— i )' + ' (y;. , -^y} .)^ (></), we 



will call the syjiuiictric dijfrroicrs of the Diatrix : 



A^A, An 



A^A.. An 



A", An. 



An. 



whereby they are defined also for the //-dimensional case. We now can 

 easily see which of the two quantities f^i and //y (the sign taken into 

 account) that is to be subtracted from the other, as we have : 



(a) d]i =_/J/ — /// if y + / is an odd number; (minuend in the 



upper half of the matrix), 

 and 



(b) du =//;• — fi I if y + / is an even number; (minuend in the 



lower half of the matrix). 



We readily see that this gives the well-known formula for Ü^;. in S3. 

 For in the case // = 3 we have : 



6) 



^2 8 = ^1 = i: ^13 = eo = j; £1. = eg = f. 



But the formula (15) also holds good in two-space. For if we have 

 ^ = Ci fi + Co f>» ^^^v must by definition in this case be a scalar: 



17) 



0.= 



1 



/11 /1. 



+ 



1 



A. A. 



-A.- A.- 



But according to (13) we must h.ave E^.^ ~ 1*, and the formula (15) 

 gives the same as (17), viz.: 



.8) 



^i\-\d,, = - (- n' + 2(y;, -y,,) =/,, -/,,. 



„The complement of the w-rowed minor (the determinant itself) is i". Bôcher, M., 

 Introduction to Higher Algebra, p. 23. 



