Page 42 - Determinants and Their Applications in Mathematical Physics

3.3 The Laplace Expansion 27

When r =2,

A n = N ir,js A ir,js , summed over i, r or j, s,

= a ij a is A ir,js .
a rj a rs
3.3.2 A Classical Proof
The following proof of the Laplace expansion formula given in (3.3.4) is
independent of Grassmann algebra.
Let

A = |a ij | n .
Then referring to the partial derivative formulas in Section 3.2.3,
∂A
= (3.3.5)
A i 1 j 1
∂a i 1 j 1
∂A i 1 j 1
= , i 1 <i 2 and j 1 <j 2 ,
A i 1 i 2 ;j 1 j 2
∂a i 2 j 2
2
∂ A
= . (3.3.6)
∂a i 1 j 1 ∂a i 2 j 2
Continuing in this way,
∂ A
r
= , (3.3.7)
A i 1 i 2 ...i r ;j 1 j 2 ...j r
∂a i 1 j 1 ∂a i 2 j 2 ··· ∂a i r j r
provided that i 1 <i 2 < ··· <i r and j 1 <j 2 < ··· <j r .
Expanding A by elements from column j 1 and their cofactors and
referring to (3.3.5),
n

A = a i 1 j 1 A i 1 j 1
i 1 =1
∂A
n

= a i 1 j 1
∂a i 1 j 1
i 1 =1
n
∂A
= a i 2 j 2 (3.3.8)
∂a i 2 j 2
i 2 =1
2
∂A ∂ A
n
= a i 2 j 2
∂a i 1 j 1 ∂a i 1 j 1 ∂a i 2 j 2
i 2 =1
n

= a i 2 j 2 A i 1 i 2 ;j 1 j 2 , i 1 <i 2 and j 1 <j 2 . (3.3.9)
i 2 =1