58] EQUIVALENCE TO MATRIC ALGEBRA 
93 
Let x be a fixed element and z, z' variable elements 
x , z z — 'Zjftiij 
of A. By (i), z — xz' is equivalent to the n equations 
which define a linear transformation T x from the initial 
variables ft, . . , . , ft t to the new variables ft 1} .... , 
ft. The determinant of T x is 
(4) 
Given the numbers ft and &{k, i = i, .... t n) of 
F such that A(x)^o, we can find unique solutions ftj 
of the n equations (3). In other words, there exists a 
unique element z' of A such that xz' = z, when 2 and x 
are given and A(x)?^o. 
Similarly, the equation z' = yz" between the foregoing 
z' and y = 'Ei7] s M s , z" = H,ft r 'u r , is equivalent to the n 
equations 
C/=i, • • • • , n), 
r, S 
which define a transformation T y from the variables 
ri y , ft to the final variables ft', ft'. 
By eliminating the ft, we get the equations of the product 
(§ 2):