Determinants and Their Applications in Mathematical Physics

248 6. Applications of Determinants in Mathematical Physics

Applying double-sum identities (C) and (A),

n

∑

r=1

n

∑

s=1

[x

r+s− 1

+(−1)

r+s

c]A

rs

=

n

∑

r=1

(2r−1)

=n

2

(6.3.12)

xA

′

A

=

n

∑

r=1

n

∑

s=1

x

r+s− 1

A

rs

=n

2

−c

n

∑

r=1

n

∑

s=1

(−1)

r+s

A

rs

. (6.3.13)

Differentiating and using (6.3.10),

(

xA

′

A

)′

=c

n

∑

r

n

∑

s

(−1)

r+s

αrαs

=cλ

2

. (6.3.14)

It follows from (6.3.5) that

xA

′

A

=

[

1 −

1

1+x

]

[n

2

−(c−1)A

11

]

=n

2

−

[

(c−1)xA

11

+n

2

1+x

]

. (6.3.15)

Hence, eliminatingxA
′
/Aand using (6.3.14),

[

(c−1)xA

11

+n

2

1+x

]′

=−cλ

2

. (6.3.16)

Differentiating (6.3.7) and using (6.3.10) and the first equation in (6.3.8),

β

′

i=λαi. (6.3.17)

Differentiating the second equation in (6.3.11) and using (6.3.17),

(xA

11

)

′′

=2cλα 1 β 1. (6.3.18)

All preparations for proving the theorem are now complete.

Put

y=4(c−1)xA

11

.

Referring to the second equation in (6.3.11),

y

′

=4(c−1)(xA

11

)

′

=4c(c−1)β

2

1. (6.3.19)

Referring to the first equation in (6.3.11),

(

y

x

)′

=4(c−1)(A

11

)

′