669 
1908-9.] Superadjugate and Skew Determinants. 
adjugate, it may be approximately spoken of as the superadjugate. It 
may also be convenient to denote it by S, and its (p,q) th element by s pq . 
3. The product of the 1C column of the superadjugate by the \d h column 
of the original determinant A is — a ]lk A if h be different from k, and is 
+ a hh A if h and k be the same. (i.) 
For 
ClTi j ^2/i ) • • ■ > $ n h 0 d'lk 3 ^2 k } • ■ • } ^nk) 
3 A 27 1 j • • • j A n ; ( ) CL j 3 • • • 3 ^ nk ) ~ ® /ifc A 
f 2$/^ • 0 CL ] t *A it h =%= h , 
\ 2ci hh . A — c^ fc A if = k . 
From this it follows that 
ffii A 
S . A = “ a2lA 
-«31 A 
- « 12 A - a 13 A . . . . 
a 22 A - a 23 A . . . . 
- a 32 A tt 33 A . . . . 
and thence we conclude that The superadjugate of any determinant A is 
equal to A n-1 A , where A is what A becomes on changing the sign of all the 
non-diagonal elements. (ii.) 
If, then, A be any skew determinant, that is to say, a skew determinant 
with any kind of diagonal whatever, A is obtainable from A by merely 
changing rows into columns. We are thus led to affirm that The super- 
adjugate of any skew determinant A is equal to A n . (iii.) 
The case of this where the diagonal elements of A are all 1 is the 
property of Cayley’s orthogonant with which we started. It may be noted 
that this particular case can also be established independently by multi- 
plying S and A together row- wise. 
4. The condition that the superadjugate may be the n th power of the 
original is seen from (ii.) to be A = A . But by expressing A as a series 
arranged according to products of the diagonal elements there is obtained 
d,y, 
+ (^11^22 ' ' ’ ^"n-2,n-‘2 
+ (^ 11^22 - ‘ ^n—o , n—3 
O'n— 1 , n—l^nn l()/ 
2 , n—‘f^n—l , n—l^nn !(jy 
+ 
+ | cl^cl^ • • • a nn |q , 
where the subscript zero is used to indicate that the determinant to which 
it is attached has had all its diagonal elements changed to 0 : and treating 
A in the same way we have 
A - 
a ii a 22 
a r . 
+ 2 
yCL\\CL‘. 
22 
a. 
a 
n — 1 
n _l CL n ( n Ifl) 
