Page 299 - Determinants and Their Applications in Mathematical Physics

284 6. Applications of Determinants in Mathematical Physics

= c r c s A rs . (6.9.13)
r s
The determinants A, B, P, and Q, their cofactors, and their complex
conjugates are related as follows:
B = Q n+1,n+2;n+1,n+2 , (6.9.14)
A = B + Q n+1,n+1 , (6.9.15)
∗
A =(−1) (B − Q n+1,n+1 ), (6.9.16)
n
P n+1,n+2 = Q n+1,n+2 , (6.9.17)
P n+1,n+2 =(−1) n+1 Q n+2,n+1 , (6.9.18)
∗
P n+2,n+2 = Q n+2,n+2 + Q, (6.9.19)
P ∗ =(−1) n+1 (Q n+2,n+2 − Q). (6.9.20)
n+2,n+2
The proof of (6.9.14) is obvious. Equation (6.9.15) can be proved as follows:

1

1

···
. (6.9.21)

B + Q n+1,n+1 = [b ij ] n ···

···

1
−1 − 1 ··· − 1 1

n+1
Note the element 1 in the bottom right-hand corner. The row operations

R = R i − R n+1 , 1 ≤ i ≤ n, (6.9.22)
i
yield

0

0

···
. (6.9.23)

···
B + Q n+1,n+1 =
[b ij +1] n

···

0
−1 − 1 ··· − 1 1

n+1
Equation (6.9.15) follows by applying (6.9.7) and expanding the determi-
nant by the single nonzero element in the last column. Equation (6.9.16) can
be proved in a similar manner. Express Q n+1,n+1 − B as a bordered deter-
minant similar to (6.9.21) but with the element 1 in the bottom right-hand
corner replaced by −1. The row operations

R = R i + R n+1 , 1 ≤ i ≤ n, (6.9.24)
i
leave a single nonzero element in the last column. The result appears after
applying the second line of (6.9.8).