Page 87 - Determinants and Their Applications in Mathematical Physics

72 4. Particular Determinants

Lemma 4.16.
E n = δ n,even .
Proof. Perform the column operation

C = C n + C 1 ,

n
expand the result by elements from the new C n , and apply Lemma 4.13
E n =(−1) n−1 B n−1 − E n−1
=1 − E n−1
=1 − (1 − E n−2 )
= E n−2 = E n−4 = E n−6 , etc.
Hence, if n is even,
E n = E 2 =1

and if n is odd,
E n = E 1 =0,
which proves the result.

Lemma 4.17. The function E ij defined in Lemma 4.15 is the cofactor of
the element ε ij in E 2n .
Proof. Let

2n

λ ij = ε ik E jk .
k=1
It is required to prove that λ ij = δ ij .
i−1
2n

λ ij = ε ik E jk +0+ ε ik E jk
k=1 k=i+1
i−1
2n

= − E jk + E jk
k=1 k=i+1
i−1

2n

= − E jk − E ji .
k=1
k=i
If i<j,
λ ij =(−1) j+1 δ i,odd − δ i,even +(−1) i
=0.
If i>j,
λ ij =(−1) j+1 δ i,even − δ i,odd − (−1) i