ioo CHARACTERISTIC, RANK EQUATIONS [chap, vn 
For, if k is any positive integer, (8) imply 
R x k=Rl, S x k=S k x . 
Multiply each member by the coefficient of u k in g{u>), 
sum as to k, and apply (9) and the similar equations in S. 
We get (17). 
First, let A have a modulus. Choose in turn as 
g(x) the characteristic determinants 8{co) and 8'(of) of 
matrices R x and S x , respectively. Then, by (17) and 
§ 59, 
R5{x) — 8(R x )=o , Ss'(x) = 8'(S x ) =0. 
Hence 8(x)=o, 8'(x)=o by Theorem 1 of § 58. 
Second, let A lack a modulus and extend it to an 
algebra A* with a modulus u 0 defined by (10). Choose in 
turn as g(x) the characteristic determinants of matrices R* x 
and Si, which by (13) are evidently equal to — co5(co) and 
— c08'(of), respectively. By the facts used in the proof 
of Theorem 3 of § 58, equations (17) hold when R and S 
are replaced by R* and S*, respectively. Hence (§ 59), 
R-xdix) — O , S- xS '(x) = O . 
Since A* has a modulus, Theorem 1 of § 58 shows that 
the subscripts are zero. 
Theorem.* For every element x of any associative 
algebra A, xS(x) =0, xS'(x) =0. If A has a modulus, also 
8(x) =0, 8'(pc) =0. 
* For another proof, with an extension to any non-associative 
algebra, see the author’s Linear Algebras (Cambridge, 1914), pp. 16-19. 
That proof is based on the useful fact that if we express xuj as a linear 
function of Ui, .... , u n and transpose, we obtain n linear homo 
geneous equations in u ly .... , u n the determinant of whose co 
efficients is 8{x). Similarly, starting with ujx we obtain d'(x). Com 
pare § 95.