Determinants and Their Applications in Mathematical Physics

4.10 Henkelians 3 133

=

cij

2 i− 1

.

After removing the factor (2i−1)
− 1
from rowi,1≤i≤n, the result is

U=

2

n

n!

(2n)!

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

x

x

3

[cij]n x

5

···

x

2 n− 1

111 ··· 1 •

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

n+1

.

Transposing,

U=

2

n

n!

(2n)!

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

1

1

[−cij]n 1

···

1

xx

3

x

5

··· x

2 n− 1

• 
  

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

n+1

.

Now, change the signs of columns 1 tonand row (n+ 1). This introduces

(n+ 1) negative signs and gives the result

U=

(−1)

n+1

2

n

n!

(2n)!

Z. (4.10.27)

Perform the column operations

C

′

j=Cj+Cn+1,^1 ≤j≤n,

onV. The result is that [aij]nis replaced by [a

∗

ij

]n, where

a

∗

ij=aij+

1

2 i− 1

.

Perform the row operations

R

′

i=Ri−

x

2 i− 1

2 i− 1

Rn+1, 1 ≤i≤n,

which results in [a
∗
ij
]nbeing replaced by [a
∗∗
ij
]n, where

a

∗∗

ij

=a

∗

ij

−

x

2(i+j−1)

2 i− 1

=

cij

2 i− 1

.

After removing the factor (2i−1)
− 1
from rowi,1≤i≤n, the result is

V=

2

n

n!

(2n)!

Z. (4.10.28)

The theorem follows from (4.10.27) and (4.10.28).