Page 325 - Determinants and Their Applications in Mathematical Physics

310 Appendix

1 2 ... (i − 1)(i +1) ... (m − 1)
i+m+1
=(−1) sgn ,
r 3 r 4 ... ... ...
m−2
r m
where 1 ≤ r k ≤ m − 2, r k = i, and 3 ≤ k ≤ m.
Proof. The cases i = 1 and i> 1 are considered separately. When i =1,
then 2 ≤ r k ≤ m − 1. Let p denote the number of inversions required to
transform the set {r 3 r 4 ...r m } m−2 into the set {23 ... (m − 1)} m−2 , that
is,

2 3 ... (m − 1)
(−1) = sgn .
p
r 3 r 4 ...
m−2
r m
Hence

1 2 3 4 ... m
sgn
i m r 3 r 4
... r m
m

1 2 3 4 ... m
=(−1) sgn
p
i m 23 ... (m − 1)
m

1234 ... (m − 1)m
p+m−2
=(−1) sgn
1234 ... (m − 1)m
m
=(−1) p+m−2

2 3 ... (m − 1)
m−2
=(−1) sgn ,
r 3 r 4 ...
m−2
r m
which proves the lemma when i =1.
When i> 1, let q denote the number of inversions required to transform
the set {r 3 r 4 ··· r m } m−2 into the set {12 ··· (i − 1)(i +1) ··· (m − 1)} m−2 .
Then,

1 2 ... (i − 1)(i +1) ... (m − 1)
(−1) = sgn .
q
r 3 r 4 ... ... ...
m−2
r m
Hence,

1 2 3 4 ··· m
sgn
i m r 3 r 4
··· r m
m

1 2 3 4 ··· ··· ··· (m − 1) m
=(−1) sgn
q
i m 12 ··· (i − 1)(i +1) ··· (m − 2) (m − 1)
m

1234 ··· ··· ··· (m − 1) m
=(−1) q+m sgn
i 123 ··· (i − 1)(i +1) ··· (m − 1) m
m

1234 ··· m
q+m+i−1
=(−1) sgn
1234 ··· m
m
=(−1) q+m+i−1

1 2 ··· (i − 1)(i +1) ··· (m − 1)
m+i−1
=(−1) sgn ,
r 3 r 4 ··· ··· ···
m−2
r m