S8] 
EQUIVALENCE TO MATRIC ALGEBRA 
95 
Similarly, from z'=z"y we obtain t y . Then z=z"q, 
q=yx. This makes it plausible that t x t y — t q . A formal 
proof follows from (2) as before. The determinant of 
(5) is denoted by A'(x). If it is not zero, there exists a 
unique element z’ such that z'x = z. 
We shall denote the matrix of transformation (3) by 
R x and that of (5) by S x , whence 
(6) R x ~ (p k j), Pkj= ijk (k,j= 1, • . . . , »), 
i= I 
having the element in the &th row and7th column; 
n 
(7) S x =*(ff k j), (Tkj= jik (k,j= I, 
i=i 
We shall call R x and S x the first and second matrices of x 
(with respect to the chosen units u l7 , u n ). 
Since the matrix of a product of two transformations is 
equal to the product of their matrices (§ 3), we have 
(8) 
RxRy=R 
xy J 
•SxR y R y x 
The determinants A(x) and A'{x) of R x and S x are 
called the first and second determinants of x (with respect 
to UiUn) • 
Since R x is the matrix of transformation (3), R x = o 
implies that is zero identically in the and hence 
that o = xz' for every z' in A. Similarly, S x = o implies 
that o = z'x for every z' in A. In particular, 
Theorem 1. If A has a modulus, either R x = o or 
S x =o implies x = o.