Determinants and Their Applications in Mathematical Physics

10 2. A Summary of Basic Determinant Theory

=

m

∑

k 1 =1

m

∑

k 2 =1

···

m

∑

kn=1

∣

∣

C 1 k 1 ···Cjkj···Cnkn

∣

∣

n

.

The function on the right is the sum ofm

n

determinants. This identity

can be expressed in the form

∣

∣

∣

∣

∣

m

∑

k=1

a

(k)

ij

∣ ∣ ∣ ∣ ∣ n

=

m

∑

k 1 ,k 2 ,...,kn=1

∣

∣a

(kj)

ij

∣

∣

n

.

h.Column Operations. The value of a determinant is unaltered by adding

to any one column a linear combination of all the other columns. Thus,

if

C

′

j

=Cj+

n

∑

r=1

krCr kj=0,

=

n

∑

r=1

krCr,kj=1,

then

∣

∣

C 1 C 2 ···C

′

j···Cn

∣

∣

=

∣

∣

C 1 C 2 ···Cj···Cn

∣

∣

.

C

′

j

should be regarded as a new columnjand will not be confused

with the derivative ofCj. The process of replacingCjbyC

′

j

is called a

column operation and is extensively applied to transform and evaluate

determinants. Row and column operations are of particular importance

in reducing the order of a determinant.

Exercise.If the determinantAn=|aij|nis rotated through 90

◦

in the

clockwise direction so thata 11 is displaced to the position (1,n),a 1 nis dis-

placed to the position (n, n), etc., and the resulting determinant is denoted

byBn=|bij|n, prove that

bij=aj,n−i

Bn=(−1)

n(n−1)/ 2

An.

2.3.2 Matrix-Type Products Related to Row and Column

Operations

The row operations

R

′

i=

3

∑

j=i

uijRj,uii=1, 1 ≤i≤3; uij=0, i>j, (2.3.1)