This result follows from application of elementary column operations on column n
of I (P-T)n and employing the definition of I (P- ) |. The essential step is noting that
the cofactor of each of the (unit) elements in column n is equal to the corresponding
Since the numerator determinant defined in Proposition 6.3 is generated from
(P I) by replacement of row m by the row vector e its value is the cofactor of the
(unit) element in row m and column n. As indicated in the following proposition, it is
equal to the determinant which results from the corresponding row replacement in (P- ).
Proposition 6.5: The numerator determinant defined in Proposition 6.3 can be written as
(P- 1 =P IP- )Fm)
where (P- I) is defined as the matrix which results from (P-I) by replacing the row
indexed by m with the row vector e.
Next, note that if m = n, then I (P- )) can be written as
(P-I) C =
where Pi is defined as the matrix which results by replacing row m of P by the row vec-
tor 0. Ifm n, then by writing the replacement row in I (P- i) (l as
-= + er -= e F
e e-. ee = e-,, e- (-ez)1,
| (P-i1) ml can be written as the difference of two determinants derived from (P-1), one
with the mh row replaced by I[- e] and the second with the mi row replaced by [- r1.
The -e- term in each row replacement provides the necessary principal diagonal contri-
bution to permit expression of I (P I) |) as
(P- )(p --n) >=(gn-_,)|-|(pW -i)|)