Determinants and Their Applications in Mathematical Physics

5.8 Some Applications of Algebraic Computing 233

and denote the result byZn+1. Verify the formula

Zn+1=−n

2

Kn(x

2

−1)

n

2

− 2

(x

2

−n

2

)

for several values ofn.

Both formulas have been proved analytically, but the details are

complicated and it has been decided to omit them.

Exercise.Show that

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

a 11 a 12 ··· a 1 n x

a 21 a 22 ··· a 2 n x

2

··· ··· ··· ··· ···

an 1 an 2 ··· ann x

n

1

x

3

···

x

n− 1

2 n− 1

−

1

2

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

=−

1

2

KnF

(

n,−n;

1

2

;−x

)

,

where

aij=

(1 +x)

i+j− 1

−x

i+j− 1

i+j− 1

and whereF(a, b;c;x) is the hypergeometric function.

5.8.7 Determinantal Identities Related to Matrix Identities

IfMr,1≤r≤s, denote matrices of ordernand

s

∑

r=1

Mr= 0 ,s> 2 ,

then, in general,

s

∑

r=1

|Mr| =0,s> 2 ,

that is, the corresponding determinantal identity isnotvalid. However,

there are nontrivial exceptions to this rule.

LetPandQdenote arbitrary matrices of ordern. Then

1. a.(PQ+QP)+(PQ−QP)− 2 PQ= 0 , alln,

b.|PQ+QP|+|PQ−QP|−| 2 PQ|=0,n=2.

2. a.(P−Q)(P+Q)−(P

2

−Q

2

)−(PQ−QP)= 0 , alln,

b.|(P−Q)(P+Q)|−|P

2

−Q

2

|−|PQ−QP|=0,n=2.

3. a.(P−Q)(P+Q)−(P

2

−Q

2

)+(PQ+QP)− 2 PQ= 0 , alln,

b.|(P−Q)(P+Q)|−|P

2

−Q

2

|+|PQ+QP|−| 2 PQ|=0,n=2.

The matrix identities 1(a), 2(a), and 3(a) are obvious. The corresponding

determinantal identities 1(b), 2(b), and 3(b) are not obvious and no neat

proofs have been found, but they can be verified manually or on a computer.

Identity 3(b) can be obtained from 1(b) and 2(b) by eliminating|PQ−QP|.