§75] DIRECT PRODUCT OF MATRIC ALGEBRAS 119 
type (1), such that each b rs is commutative with every 
&ij and such that the m 2 n 2 products dijh rs are linearly 
independent with respect to F. 
Then those products are the basal units of the direct 
product A XB (§ 50). Take them as the elements of a 
matrix (e Pq ) which is exhibited compactly as the com 
pound matrix 
• • • ifltf)bln 
in which the entries themselves are matrices : 
From our two notations for the same element, we 
have 
P dijb rs , 
Q = dklb tu =ek+ m {t-i), l+m{u—i) • 
Evidently PQ = o unless k=j, t = s, and then 
But k=j, i = s imply j+m(s — i) —1) and con 
versely, since j and k are positive integers Sm. Hence 
the e’s satisfy relations of type (1) and are therefore the 
basal units of a simple matric algebra. 
Theorem. The direct product of two simple matric 
algebras of orders m 2 and n 2 is a simple matric algebra of 
order m 2 n 2 .