from p by adding 1/L times the column nA e SA' {IA} to each of the L adjacent
nonabsorbing columns n e S(nA)' and repeating the process for each remaining
nA e SA' {mA}. This operation is applied once each for the exactly N 1 absorbing state
columns nA e SA' {mA} and it preserves the value of the determinant Qi^ = P^-I
If now Q~A(m I n) denotes the general element of QA then by applying the recipe
used in its construction and Eq. 7.5, the elements in the absorbing state rows
nA E SA' {mA} of Qii can be written as
-MLao + O(a2)
(I+c)ML m = nA
(1 + a) A
O(a2)
VmA e SA',V'A A SA' -{mA} Q(m I nA) -ML me S(nA)'
O(acs
(a)L m e S'- S(nA)'- {nA}
(1 + a)ML
where as before s is an integer satisfying s 2 2. Thus, each of the N 1 absorbing state
rows nA e SA' {mA} of QA I can be written as a sum of two rows, one row containing
-ML(/(l + ca)L at its principal diagonal location and zeros elsewhere and the second row
being a multiple of a2/(1 + a)ML. It follows from elementary determinant row expansion
operations that P -I = |Ai can be written as
S -(_MLaO)N-I O(as)
Pi- = I = A = + Eq. 7.6
A(1 + ML( 1) ( +()MLN'
where Q iA'A is the order N' N + 1 principal minor of Qmi generated by deleting the
N 1 row/column pairs which intersect on the nA e SA' {mA} principal diagonals and
where the exponent s of a in the Eq. 7.6 order expression is an integer satisfying s 2 N.
The elements in all rows of QA' I except row m, are composed of contributions
from the elements in nonabsorbing state rows of P and the -1 principal diagonal term
contributed by I in Pj^- I. Row mA of IQA' contains -1 at its principal diagonal loca-
tion and zeros elsewhere. Thus, if QA^ I is written as A = PA I' then from the