1906 - 7 .] Dr Muir on Minors of a Product-Determinant. 
79 
XI. — The Minors of a Product-Determinant. 
By Thomas Muir, LL.D. 
(MS. received November 12, 1906. Read January 21, 1907.) 
1. If f or aa + b(3 + cy + . . . we put 
a, b , c, .... a b c . . . 
a, , y , . . . . a ft y . . . ’ 
then the product of | afb 2 c z | and \f x g 2 h s 1 ma y be written 
a l a 2 a z a Y a 2 a z a x a 2 
f\ 9\ f<2, 9 h 2 f 3 9z ^3 
b 1 b 2 & 3 \ b 2 b z b Y b 2 b z 
fi 9] K / 2 92 \ /s 9z K 
fi 9\ K U 92 h 2 fs 9z K 
and the minor which is the cofactor of the element in the place 1, 1 is 
* 
b\ b 2 b z 
f 2 f 2 
C 1 C 2 C 3 
92 9s 
h 2 h B 
1 + 1 &1 C 3 1 
•1 J 
2 b'z 1 + 
so that if, as usual, we denote the cofactors of the elements in \a 1 b 2 c s \ , 
I I b y the corresponding capital letters, we may assert that In the 
determinant which is the product of | a x b 2 c s | and ! f x g 2 h z j the cofactor of 
d- L a 2 (% 3 . A 1 A 2 A 3 
Affih Fr^H,* 
2. Proceeding to the case where the number of determinants to be 
multiplied together is three, namely 
I a i h 2 C 2 I J l/l 92 K\A m i n 2 r 3 I > 
let us denote 
a , a n a 0 
" V ( it _j_ a l a 2 a 2 r 
fi 9i K " Vl ' f 2 g 2 h 2 ' 1 / 3 g z h z 1 
* This is usually written 
I 6i &2 ^3 1 • I A A I 
I C 1 C 2 C 3 I fz 9z h 3 I , 
but throughout the present paper row-by-column multiplication is used, so as to have the 
results immediately applicable to matrices. 
