Determinants and Their Applications in Mathematical Physics

2.3 Elementary Formulas 9

Example.

∣

∣

C 1 C 3 C 4 C 2

∣

∣

=−

∣

∣

C 1 C 2 C 4 C 3

∣

∣

=

∣

∣

C 1 C 2 C 3 C 4

∣

∣

.

Applying this property repeatedly,

i.

∣

∣C

mCm+1···CnC 1 C 2 ···Cm− 1

∣

∣=(−1)(m−1)(n−1)A,

1 <m<n.

The columns in the determinant on the left are a cyclic permutation

of those inA.

ii.

∣

∣

CnCn− 1 Cn− 2 ···C 2 C 1

∣

∣

=(−1)

n(n−1)/ 2

A.

d.Any determinant which contains two or more identical columns is zero.

∣

∣

C 1 ···Cj···Cj···Cn

∣

∣

=0.

e. If every element in any one column ofAis multiplied by a scalarkand

the resulting determinant is denoted byB, thenB=kA.

B=

∣

∣

C 1 C 2 ···(kCj)···Cn

∣

∣

=kA.

Applying this property repeatedly,

|kaij|n=

∣

∣

(kC 1 )(kC 2 )(kC 3 )···(kCn)

∣

∣

=k

n

|aij|n.

This formula contrasts with the corresponding matrix formula, namely

[kaij]n=k[aij]n.

Other formulas of a similar nature include the following:

i.|(−1)

i+j

aij|n=|aij|n,

ii.|iaij|n=|jaij|n=n!|aij|n,

iii.|x

i+j−r

aij|n=x

n(n+1−r)

|aij|n.

f.Any determinant in which one column is a scalar multiple of another

column is zero.

∣

∣

C 1 ···Cj···(kCj)···Cn

∣

∣

=0.

g.If any one column of a determinant consists of a sum ofmsubcolumns,

then the determinant can be expressed as the sum ofmdeterminants,

each of which contains one of the subcolumns.

∣

∣

∣

∣

∣

C 1 ···

(

m

∑

s=1

Cjs

)

···Cn

∣

∣

∣

∣

∣

=

m

∑

s=1

∣

∣C

1 ···Cjs···Cn

∣

∣.

Applying this property repeatedly,

∣

∣

∣

∣

∣

(

m

∑

s=1

C 1 s

)

···

(

m

∑

s=1

Cjs

)

···

(

m

∑

s=1

Cns

)∣

∣

∣

∣

∣