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)