Determinant Notation. 

 may be denoted by 



M 



:vj7 



,p,q=l,. ..m\ 

 \r, 8 = 1 , . . . n / 



As an example take the theorem given long ago by 

 Sylvester*, as an illustration of the power of this umbral 

 notation. 



In the notation now suggested that theorem may be 

 expressed as follows : — 



If 



then 



(a pq ) {b ps ) 



\Crq) "rs 



(p,q = l,..m); 



7'2 



(Crq) (drs) 



{r,s=l, ... n). 



Proofs of this theorem have been given by Scott f, Fro- 

 benius J, and Netto §. But in no one of these proofs is the 

 fundamental simplicity of the theorem made evident. 



In the first place, by the general law of extension due to 

 Muir ||, the theorem is at once seen to be true, because it is 

 merely the extensional of the obvious identity 



d rs I = 



(rfn) (d 12 ) 



{din) 



(d 2n ) 



(d n i) {d n2 ) . . . {d nn ) 



In the second place, the theorem may be proved directly 

 by the following elementary method. 



Multiply columns 1 to m of the determinant A, where 



A = 



{<tpq) {bps) 

 (frq) {drs) 



/p,g = l,...m\ 

 \r,s = l,.. .n), 



by \„, . . . \ sm , add to column m + s, and do this for all values 

 of s. Then, provided the multipliers \ are taken so that 



2*q Apq ">sq T Op S — U, 



(1) 



* Phil. Mag. April 1851, p. 297. 



t Proc. Lond. Math. Soc. vol. xiv. p. 92. 



\ Crel/e, vol. cxiv. p. 189. 



§ Acta Math. vol. xvii. p. 202. 



|| Trans. R. S. E. xxx. p. 1 ; ' Theory of Determinants,' p. 213. 



