357] A SUPPLEMENTARY MEMOIR ON THE THEORY OF MATRICES. 
439 
which exhibits very clearly the terms which are to be combined together; thus in 
the upper left-hand corner we have (a, b, c][a, a', a"), and so for the other places in 
the compound matrix. 
2. It is not in the Memoir on Matrices explicitly remarked, but it is easy to 
see that sums of matrices, all the matrices being of the same order, may be multiplied 
together by the ordinary rule; thus 
(A+B) (G + D) = AC + AD + BG + BD: 
this remark will be useful in the sequel. 
Article Nos. 3 to 13. First Investigation. 
3. We have to consider the formulas for the automorphic linear transformation 
of the function xw' 4- yz — zy' — wx', that is, of the function 
( 0, 0, 0,-1 $£, y, 0, w\x, y', z\ w') 
0, 0,-1, 0 
1, 0, 0, 0 
0, 1, 0, 0 
= (Il\x, y, z, w^x', y, z\ w'\ 
viz., if the variables are transformed by the formulas 
(x, V, z, w) = (n£X, F, W), 
(x\ y', z', w') = (n$X', F, ¿7, W), 
then the matrix (II) is such that we have identically 
(OJ», y> z, y, z\ w') = (n][X, F, Z, F$X', F, Z\ W); 
the expression for (II) is given in my memoir [153] above referred to; viz. observing 
that the matrix (O) is skew symmetrical, then (No. 13) we have 
n = il" 1 (Q — T) (O + T)" 1 B, 
where T is an arbitrary symmetrical matrix. 
4. I propose to compare with the matrix II the inverse matrix II -1 . Recollecting 
that in the theory of matrices (ABGD) -1 = D -1 (7 _1 B~ x A -1 , we have 
n-1 = ft- 1 (ft + T) (fl — T)- 1 a ; 
and it is to be shown that II and n _1 are composed of terms which (except as to 
their signs) are the same in each, so that either of these matrices is derivable from 
the other by a peculiar form of transposition. It is to be borne in mind throughout 
that T is symmetrical, il skew symmetrical.