98 CHARACTERISTIC, RANK EQUATIONS [chap, vii 
Note that (13) is obtained by bordering matrix R x 
in (6) with a front column of £’s and then a top row of 
zeros. Write x'= ^jUj. Then 
XX = IjpkUk , pk Pkfij • 
j 
We verify at once that the product R*R* is R* x r since 
it is obtained by bordering matrix R XX ’ = R X R X ' with a 
front column of p’s and a top row of zeros. Again, 
(9) imply the corresponding equations in R*, 
Theorem 3. Any associative algebra A {without a 
modulus) is equivalent to the algebra whose elements are 
the bordered first matrices (13) of the elements x of A, and is 
reciprocal to the algebra whose elements are the bordered 
second matrices S* of the elements x of A. 
Here S* is obtained by bordering matrix S x with a 
front column of £’s and a top row of zeros, and hence 
may be derived from (13) by replacing each pkj by a k j. 
Theorem 4. Every transformation T x is commutative 
with every transformation t y . Hence 
(14) R^Sy—SyRx 
for all elements x and y of A if and only if A is associative. 
For, if we apply first transformation z = xz' and 
afterward transformation z' = z"y, we obtain 
T x t y \ z=X'z"y. 
But if we apply first t y : z = z'y and afterward T x : z' = 
xz", we get 
t y T x : z=xz” • y. 
The group of the transformations T x and the group 
of t y are said to be a pair of reciprocal groups in Lie’s