Determinants and Their Applications in Mathematical Physics

72 4. Particular Determinants

Lemma 4.16.

En=δn,even.

Proof. Perform the column operation

C

′

n=Cn+C^1 ,

expand the result by elements from the newCn, and apply Lemma 4.13

En=(−1)

n− 1

Bn− 1 −En− 1

=1−En− 1

=1−(1−En− 2 )

=En− 2 =En− 4 =En− 6 ,etc.

Hence, ifnis even,

En=E 2 =1

and ifnis odd,

En=E 1 =0,

which proves the result.

Lemma 4.17. The functionEijdefined in Lemma 4.15 is the cofactor of

the elementεijinE 2 n.

Proof. Let

λij=

2 n

∑

k=1

εikEjk.

It is required to prove thatλij=δij.

λij=

i− 1

∑

k=1

εikEjk+0+

2 n

∑

k=i+1

εikEjk

=−

i− 1

∑

k=1

Ejk+

2 n

∑

k=i+1

Ejk

=

[

2 n

∑

k=i

−

i− 1

∑

k=1

]

Ejk−Eji.

Ifi<j,

λij=(−1)

j+1

[

δi,odd−δi,even+(−1)

i

]

=0.

Ifi>j,

λij=(−1)

j+1

[

δi,even−δi,odd−(−1)

i

]