Determinants and Their Applications in Mathematical Physics

3.3 The Laplace Expansion 27

Whenr=2,

An=

∑

Nir,jsAir,js, summed overi,rorj, s,

=

∑

∣

∣

∣

∣

aij ais

arj ars

∣

∣

∣

∣

Air,js.

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=|aij|n.

Then referring to the partial derivative formulas in Section 3.2.3,

Ai

1 j 1

=

∂A

∂ai 1 j 1

(3.3.5)

Ai 1 i 2 ;j 1 j 2 =

∂Ai

1 j 1

∂ai 2 j 2

,i 1 <i 2 andj 1 <j 2 ,

=

∂

2

A

∂ai 1 j 1 ∂ai 2 j 2

. (3.3.6)

Continuing in this way,

Ai 1 i 2 ...ir;j 1 j 2 ...jr=

∂

r

A

∂ai 1 j 1 ∂ai 2 j 2 ···∂airjr

, (3.3.7)

provided thati 1 <i 2 <···<irandj 1 <j 2 <···<jr.

Expanding A by elements from column j 1 and their cofactors and

referring to (3.3.5),

A=

n

∑

i 1 =1

ai

1 j 1

Ai

1 j 1

=

n

∑

i 1 =1

ai 1 j 1

∂A

∂ai 1 j 1

=

n

∑

i 2 =1

ai

2 j 2

∂A

∂ai

2 j 2

(3.3.8)

∂A

∂ai 1 j 1

=

n

∑

i 2 =1

ai 2 j 2

∂

2

A

∂ai 1 j 1 ∂ai 2 j 2

=

n

∑

i 2 =1

ai

2 j 2

Ai

1 i 2 ;j 1 j 2

,i 1 <i 2 andj 1 <j 2. (3.3.9)