A Determinantal Identity - Nonlinear Oligopolies: Stability and Bifurcations - page 307

where all other matrix elements are zeros. Expanding this determinant with respect
to its last column we obtain a recursive relation

a 2
a 1

a 3
a 2

a N
a N 1

D D N 1 1 C . 1/ N 1 a 1 b N .

/. /:::.

D D N 1 C a N b N

D N /

completing the proof. t

Two particular applications of identity (E.1) will be shown next.
Consider first the determinant with the simple structure

0
@

1
A

A 1 ./ B 1 ./ ::: B 1 ./
B 2 ./ A 2 ./ ::: B 2 ./
: : : : : : : : :

Det D det : (E.2)

: : :
B N ./ B N ./:::A N ./

By introducing vectors b ./ D .B 1 ./;:::;B N .// T ; 1 T D .1;:::;1/ and the

diagonal matrix D ./ D diag.A 1 ./ B 1 ./;:::;A N ./ B N .// we can rewrite
the determinant as

det . D ./ C b ./ 1 T / D det . D ./ det . I C D 1 ./ b ./ 1 T /:

The first determinant is diagonal, the second has the special structure of (E.1) with
a D D 1 ./ b ./ and b

T

D 1 T . So by using identity (E.1) we have

" 1 C # :

Y

N

X

N

B k ./
A k ./ B k ./

Det D .A k ./ B k .// (E.3)

k D 1 k D 1

Consider next a special matrix

0 1

a 1 b 1 ::: b 1
b 2 a 2 ::: b 2
: : : : : : : : :
b N b N :::a N

@ A

A D : (E.4)

The characteristic polynomial of this matrix can be determined by using relation
(E.3). Notice that

0
@

1
A

a 1 1 ::: b 1
b 2 a 2 ::: b 2
: : :

det . A I / D det ;

: : : : : :

b N b N :::a N