284 6. Applications of Determinants in Mathematical Physics
=
∑
r∑
scrcsArs. (6.9.13)The determinantsA,B,P, andQ, their cofactors, and their complex
conjugates are related as follows:
B=Qn+1,n+2;n+1,n+2, (6.9.14)A=B+Qn+1,n+1, (6.9.15)A
∗
=(−1)n
(B−Qn+1,n+1), (6.9.16)Pn+1,n+2=Qn+1,n+2, (6.9.17)P
∗
n+1,n+2=(−1)n+1
Qn+2,n+1, (6.9.18)Pn+2,n+2=Qn+2,n+2+Q, (6.9.19)P
∗
n+2,n+2=(−1)n+1
(Qn+2,n+2−Q). (6.9.20)The proof of (6.9.14) is obvious. Equation (6.9.15) can be proved as follows:
B+Qn+1,n+1=∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
1
1
···
[bij]n ······1− 1 − 1 ··· − 11∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
n+1. (6.9.21)
Note the element 1 in the bottom right-hand corner. The row operations
R
′
i=Ri−Rn+1,^1 ≤i≤n, (6.9.22)yield
B+Qn+1,n+1=∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
∣
∣
∣
∣
∣
0
0
···
[bij+1]n ······0− 1 − 1 ··· − 11∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
∣
∣
∣
∣
∣
n+1. (6.9.23)
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. ExpressQn+1,n+1−Bas 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
′
i=Ri+Rn+1,^1 ≤i≤n, (6.9.24)leave a single nonzero element in the last column. The result appears after
applying the second line of (6.9.8).