﻿538 Dr. T. Muir on the Expressibility of a 



Next it is clear tbat 



x 1? x 2 , x 3 , . . , x n may be so chosen that all the elements of any 

 one of the rows or columns, except the diagonal element, shall 

 be 1. 



For example, the first row, 



x x x } x 1 



«11> #12 — ? #13— > • • • > a l»~ 

 X 2 «i3 X n 



may be made to take the form 



a n , 1, 1, . . . , 1 



by giving x h x 2 , x 3 , . . ., x n the values 1, a u , a u , . . . , a,„ 

 respectively. 



It follows, therefore, that 



Any determinant of the n th order may be transformed so as 

 to have lfor n— 1 o/ its elements, and yet the determinant itself 

 and all its coaxial minors remain unaltered in value. 



3. This is the same as saying that 2 ra — 1 quantities, viz. 

 the determinant and its coaxial minors, can be expressed in 

 terms of n 2 — (n— 1) others, viz. the modified elements, which 

 are not equal to unity. Eliminating the latter, and we have 

 2 W — ri 2 -¥n — 2 relations connecting the former — and this is 

 Major MacMahon's auxiliary theorem. 



4. The proof leaves no doubt as to the existence of two rela- 

 tions between any determinant, when of even order higher 

 than the second, and its coaxial minors : but it is worth while 

 to intensify the conviction by putting the relations actually 

 in evidence for a particular determinant. Perhaps the deter- 

 minant which lends itself most easily to the end in view is 



Here the four coaxial minors of the 3rd order are 



1 



1 



1 





1 



1 

 1 



a 





1 





1 







a 





' 



1 1 



1 





1 1 



b 





»i 



1 



> 



1 1 1 



lie 



1 - 1 1 



1 

 1 



a 



1 1 

 b c 



a b 

 1 c 



