Page 33 - Determinants and Their Applications in Mathematical Physics

18 3. Intermediate Determinant Theory

Exercises
1. Let δ denote an operator which, when applied to C j , has the effect
r
of dislocating the elements r positions downward in a cyclic manner
so that the lowest set of r elements are expelled from the bottom and
reappear at the top without change of order.

,
T
r
δ C j = a n−r+1,j a n−r+2,j ··· a nj a 1j a 2j ··· a n−r,j
1 ≤ r ≤ n − 1,
0
δ C j = δ C j = C j .
n
Prove that

n

0, 1 ≤ r ≤ n − 1
r
C 1 ··· δ C j ··· C n =

nA, r =0,n.
j=1
2. Prove that
n

r

C 1 ··· δ C j ··· C n = s j S j ,
r=1
where
n

s j = a ij ,
i=1
n

S j = A ij .
i=1
Hence, prove that an arbitrary determinant A n = |a ij | n can be
expressed in the form
1
n
A n = s j S j . (Trahan)
n
j=1
3.2 Second and Higher Minors and Cofactors
3.2.1 Rejecter and Retainer Minors
It is required to generalize the concept of first minors as defined in
Chapter 1.
Let A n = |a ij | n , and let {i s } and {j s },1 ≤ s ≤ r ≤ n, denote two
independent sets of r distinct numbers, 1 ≤ i s and j s ≤ n. Now let
(n)
M denote the subdeterminant of order (n − r) which is ob-
i 1 i 2 ...i r ;j 1 j 2 ...j r
tained from A n by rejecting rows i 1 ,i 2 ,...,i r and columns j 1 ,j 2 ,...,j r .
(n)
M is known as an rth minor of A n . It may conveniently be
i 1 i 2 ...i r ;j 1 j 2 ...j r