Page 40 - Determinants and Their Applications in Mathematical Physics

3.3 The Laplace Expansion 25

is applied in Section 3.6.2 on the Jacobi identity. Formulas (3.2.16) and
(3.2.17) are applied in Section 5.4.1 on the Matsuno determinant.

3.3 The Laplace Expansion

3.3.1 A Grassmann Proof

The following analysis applies Grassmann algebra and is similar in nature
to that applied in the definition of a determinant.
Let i s and j s ,1 ≤ s ≤ r, r ≤ n, denote r integers such that

1 ≤ i 1 <i 2 < ··· <i r ≤ n,
1 ≤ j 1 <j 2 < ··· <j r ≤ n
and let

n

x i = a ij e k , 1 ≤ i ≤ n,
k=1
r

e ,
y i = a ij t j t 1 ≤ i ≤ n,
t=1
z i = x i − y i .
Then, any vector product is which the number of y’s is greater than r or
the number of z’s is greater than (n − r) is zero.
Hence,
x 1 ··· x n =(y 1 + z 1 )(y 2 + z 2 ) ··· (y n + z n )

= z 1 ··· y i 1 ··· y i 2 ··· y i r ··· z n , (3.3.1)
i 1 ...i r
where the vector product on the right is obtained from (z 1 ··· z n ) by replac-

,1 ≤ s ≤ r, and the sum extends over all n combinations of
ing z i s by y i s
r
the numbers 1, 2,...,n taken r at a time. The y’s in the vector product can
be separated from the z’s by making a suitable sequence of interchanges
and applying Identity (ii). The result is
∗

p
z 1 ··· y i 1 ··· y i 2 ··· y i r ··· z n =(−1) y i 1 ··· y i r z 1 ··· z n , (3.3.2)
where
n
1
p = i s − r(r + 1) (3.3.3)
2
s=1
and the symbol ∗ denotes that those vectors with suffixes i 1 ,i 2 ,...,i r are
omitted.