Determinants and Their Applications in Mathematical Physics

3 Intermediate Determinant Theory

3.1 Cyclic Dislocations and Generalizations...........

Define column vectorsCjandC
∗
j
as follows:

Cj=

[

a 1 ja 2 ja 3 j···anj

]T

C

∗

j=

[

a

∗

1 ja

∗

2 ja

∗

3 j···a

∗

nj

]T

where

a

∗

ij=

n

∑

r=1

(1−δir)λirarj,

that is, the elementa
∗
ij
inC
∗
j
is a linear combination of all the elements

inCjexceptaij, the coefficientsλirbeing independent ofjbut otherwise

arbitrary.

Theorem 3.1.

n

∑

j=1

∣

∣

C 1 C 2 ···C

∗

j···Cn

∣

∣

=0.

Proof.

∣

∣C

1 C 2 ···C

∗

j

···Cn

∣

∣=

n

∑

i=1

a

∗

ij

Aij

=

n

∑

i=1

Aij

n

∑

r=1

(1−δir)λirarj.