Determinants and Their Applications in Mathematical Physics

140 4. Particular Determinants

=Kn

(

K

− 1

n

Qn−t

2

In

)

=Kn

(

S

2

n

−t

2

In

)

=Kn(Sn+tIn)(Sn−tIn)

=KnHnHn. 

Corollary.

B

− 1

n =H

− 1

n H

− 1

n K

− 1

n,

[

B

(n)

ji

]

=

[

H

(n)

ji

][

H

(n)

ji

][

K

(n)

ji

]

.

Lemma.

n

∑

i=1

h

(n)

ij

=x

2 j− 1

+t.

The proof applies (4.11.3) and is elementary.

LetEn+1denote the determinant of order (n+ 1) obtained by bordering

Hnas follows:

En+1=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

h 11 h 12 ··· h 1 n vn 1 /n

h 21 h 22 ··· h 2 n vn 2 /(n+1)

..................................

hn 1 hn 2 ··· hnn vnn/(2n−1)

11 ··· 1 •

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

n+1

=−

n

∑

r=1

n

∑

s=1

vnrHrs

n+r− 1

. (4.11.16)

Theorem 4.45.

En+1=(−1)

n

Hn− 1.

The proof consists of a sequence of row and column operations.

Proof. Perform the column operation

C

′

n

=Cn−x

2 n− 1

Cn+1 (4.11.17)

and apply (6b) withj=n. The result is

En+1=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

h 11 h 12 ··· h 1 ,n− 1 • vn 1 /n

h 21 h 22 ··· h 2 ,n− 1 • vn 2 /(n+1)

........................................

hn 1 hn 2 ··· hn,n− 1 tvnn/(2n−1)

11 ··· 11 •

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

n+1

. (4.11.18)

Remove the element in position (n, n) by performing the row operation

R

′

n

=Rn−tRn+1. (4.11.19)