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