Page 24 - 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.
(m−1)(n−1)
C m C m+1 ··· C n C 1 C 2 ··· C m−1 =(−1) A,

1 <m<n.
The columns in the determinant on the left are a cyclic permutation
of those in A.
n(n−1)/2
ii. C n C n−1 C n−2 ··· C 2 C 1 =(−1) A.

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

C 1 ··· C j ··· C j ··· C n =0.

e. If every element in any one column of A is multiplied by a scalar k and
the resulting determinant is denoted by B, then B = kA.

B = C 1 C 2 ··· (kC j ) ··· C n = kA.

Applying this property repeatedly,

|ka ij | n = (kC 1 )(kC 2 )(kC 3 ) ··· (kC n )

= k |a ij | n .
n
This formula contrasts with the corresponding matrix formula, namely
[ka ij ] n = k[a ij ] n .
Other formulas of a similar nature include the following:
i. |(−1) i+j a ij | n = |a ij | n ,
ii. |ia ij | n = |ja ij | n = n!|a ij | n ,
iii. |x i+j−r a ij | n = x n(n+1−r) |a ij | n .
f. Any determinant in which one column is a scalar multiple of another
column is zero.

C 1 ··· C j ··· (kC j ) ··· C n =0.

g. If any one column of a determinant consists of a sum of m subcolumns,
then the determinant can be expressed as the sum of m determinants,
each of which contains one of the subcolumns.

m m

C 1 ··· C js ··· C n = C 1 ··· C js ··· C n .

s=1 s=1
Applying this property repeatedly,

m m m

··· ···
C 1s C js C ns

s=1 s=1 s=1