(IP-II I P- )-r =IP- I I P-IIl
= (P- i)lr -(Pi,- I)P Eq. 9.12
aIp_?,I I~ _?-i.
Eq. 9.4 can be generalized to express the value of (I P- I PW-II )' as indicated
in the following. Let r have k < N' 1 rows which specify nonzero order differentiation,
let K = {n,, n2, -, nk c S' be the set of differentiated-row indices and further let m e K.
Also, for N' > u 2 k let Au(r) be the sum of all order u principal minors of I P0 formed by
including the k differentiated rows indicated by n e K and u k of the N' k undifferen-
tiated rows in | P. Exactly
N' k (N' k)!
lu-k (u k)!(N' u)!
order u principal minors are summed to produce A,(r). Finally, let Au(r) be defined simi-
larly for I Pl. Then, applying the same elementary determinant expansion rules that lead
to Eq. 9.1 and Eq. 9.3 to I P -Il and I P x-I yields
(_N'
;P X_ (-1)uA(r)XN'-u
u=k
and
-N= ) (_ ,, N- r). ,
-0 1 K-iX'-u,
u=k
and substituting these results into Eq. 9.12 with X = 1 yields the differentiated analog of
Eq. 9.4
(IP-II -IP-,I|) =|P-IP PI Eq.9.13
N'
= (-I)N -(Au(r)-Au(r)).
u=k
If 1 P K is the order k principal minor of I PMI uniquely defined by the set of
row/column indices K = {n,,n2, -,nk} cS' where m e K and if I PO % is the order k + 1
principal minor generated by including the undifferentiated row n e S' K with K (i.e.