OF THE RECIPROCAL OF A DETERMINANT. 



625 



u 2 - b.yX 2 



-b 3 x 3 



~ V C 4 





ii - b.,:c., 



- &4*4 



- Vs 





u. 2 - 6^3 



~ h X 2 



~ ] h x i 



- c. 2 x. 2 



u 3 — c 3 x 3 



- V* 



- r.,x. 2 



ti 3 c 4 u ^ 



- c 3 x 3 





~ r z x s 



U 3 ~ C 2 X 2 



-c 4 x 4 



— d 2 x 2 



-d 3 x 3 



u 4 - d 4 x 4 



> , - d.^Xt, 



-d 4 x 4 



ti/A " _ Ct-Qlt/O 



> 



— d 3 x 3 



- d. 2 x 2 



u 4 - d 4 x 4 



u 2 - \x 3 



-b 4 x 4 



- b,x 2 





u. 2 - b 4 x 4 



-b 2 x. 2 



-b 3 x 3 





u. 2 - b 4 x 4 



- b s»s 



- b. 2 x 2 



-C 3 X 3 



u 3 — c 4 x 4 



~ C 2 X 2 





- C 4 r 3 



U 3 ~ C 2 X 2 



— c 3 x 3 





~ r i x i 



U 3 ~ C 3 X 3 



~ C 2 X -2 



- d 3 x 3 



-d 4 x 4 



c4. ™" (I/ijJCn 



i 



- d 4 x 4 



-d 2 x 2 



u 4 - d 3 x 3 



> 



-d 4 x 4 



— d 3 x 3 



u 4 - d 2 x. 2 



is presentable in the form x 1 '¥ 1 , where F x is a complete quadric function of 

 x 1 , x 2 , x 3 , x 4 , consisting of sixteen positive terms, the coefficient of x r 2 being monomial 

 and that of x,.x s binomial. 



That x x is a factor is readily seen by adding the elements of each row ; thus, in 

 the case of the first determinant of the six we have at once 



<h 



b 3 x 3 



-d 3 x 3 



b 4 x 4 



d l x l + d, 2 x., + d 3 x 3 



But here the cofactor of \ consists of eight positive terms of the kind promised, the 

 two + c^d^x^ , — c i d 3 x i x 3 cancelling each other : the cofactor of c 1 has four of 

 similar kind, and the cofactor d 1 also four. The result consequently can be written 



*1 



•'■■_' 



x 3 



■'4 



V-'i^i 



b x c 2 d x 



Vl^3 



h \ r A 



Vy'-j 



^1 C 2^2 



b 1 c 2 d. 2 



b \ e i d 2 



Vi'A 



h c \ d 2 



h'-i'h 



b i'h d 3 



Vi^i 



b 4 c. 2 d x 



6 s c 4 rfj 



b 4 c 4 d x 



(18) If the elements of each row in the quadrate array here obtained be divided by 

 the diagonal element of that row, there results the more interesting form 



x l 



ar 2 



*3 



x 4 



1 



£2 



d 3 



'■4 





c i 



d, 



'■l 



c ± 



1 



'Is 



£4 



Co 





d. 2 



C 2 



d\ 



d. 2 



1 



K 



d Z 



<h 



h 



C l 



Y 



h 



1 



C 4 



''4 



h 





X l 



/ VY / ] 



x 2 



b 1 c 2 d. 2 



X 3 



h c \ ( h 



x 4 



b 4 c 4 d l 



of which the quadrate array is inverso-symmetric* 



The existence of this peculiar symmetry gives us the means of deriving the twelve 

 non-diagonal elements from those in the diagonal ; and as the latter four are easily 

 obtained we can thus write out the full expansion without any delay upon intermediate 

 work. Let us take as an example the third of the six determinants, viz., 



Vs 



~ fe 2*2 



-b 4 x 4 



c 3 x 3 



u 3 — c.,.r 



- C484 



d 3 x 3 



-djc 2 



u 4 - d 4 x 4 



* The first to draw attention to determinants of such array was probably Joseph Horner : see his " Notes on 

 Determinants" in the Quart. Journ. of Math., viii. pp. 157-162 (year 1865). 



