(293) 
SECOND NOTE ON THE DETERMINANT OF THE SUM OF 
TWO CIECULANT MATRICES. 
By Sir Thomas Muir, LL.D. 
(1) Almost forty years ago the curious identity 
1 
ao + fcg 
tig + 63 
% + ?>5 
1 
h 
«4 
-h 
1 
«2 + ^4 
«8 + ^5 
a^ + Z/j 
1 
-h 
h 
«'3 
-h 
«4 
I 
% + &4 
+ 
+ ho 
1 
-h 
«1- 
h 
^2 
-h 
0.3 
1 
^^4 + ^5 
«5 + ^l 
+ ^3 
«o + ^3 
1 
-h 
%>~ 
h 
«1 
-\ 
«2 
-h 
1 
«3 + ^l 
1 
«8 
-\ 
h 
-h 
was pointedly drawn attention to* in the hope that a purely determinant 
proof might be forthcoming. In the long-continued absence of such a proof 
I propose to supply one, not so much, however, on account of the importance 
of the identity itself as of the incidental and subsequent theorems to which 
the attempt has led up. 
(2) At the outset it is clear that each of the two determinants involved is 
expressible as a sum of sixteen (2*) determinants with monomial elements : 
and, further, that as they differ only in the signs of the h's, a number of the 
one set of sixteen must cancel the same number of the other set. For short- 
ness' sake and definiteness of statement let us denote the sixteen on the left, 
namely, 
1 a~ 
1 
1 a- ctj ^2 % 
1 (X4 
1 ^3 «5 
by 
101111 I, 101112 I, 
the constant column being indicated by 0, a column of a's by 1, and a column 
of 6's by 2. We then see that, if in the symbol for a determinant on the 
right there be an even number of 2's, the said determinant cannot differ from 
the corresponding determinant on the left ; and if there be an odd number 
of 2's the difference existing is merely a difference of sign. It thus follows 
by subtraction that what we are reduced to showing is that the double of 
the sum of the determinants on the right that have an odd number of 2's is 
equal to 0 : that is, that 
2{|01112| + |01121| + |01211|-f|01222| 
-h|02111| + |02122| + |02212|+|02221|}=0. 
* ' Analyst/ x (1882), pp. 8-9. 
23§ 
1 «2 ^4 &5 
1 ai ao «3 
1 a- 
1 b^ 
1 ao a, a- b,, , 
