Determinants and Their Applications in Mathematical Physics

5.6 Distinct Matrices with Nondistinct Determinants 211

=

1

Bn

r

∑

t=0

cr−tB

(n+1)

n+1,n+1−t.

Hence, from the fourth equation in (5.5.11) and applying Lemma (b) and

(5.5.31),

BnQ 2 n=

n

∑

r=0

x

r

r

∑

t=0

cr−tB

(n+1)

n+1,n+1−t

=

n

∑

j=0

B

(n+1)

n+1,j+1

j

∑

r=0

crx

n+r−j

=

n

∑

j=0

ψjx

n−j

B

(n+1)

n+1,j+1

This sum is an expansion of the determinant in part (b) of the theorem.

This completes the proofs of both parts of the theorem.

Exercise.Show that the equations

hn, 2 n+j=0,j≥ 2 ,

kn, 2 n+j=0,j≥ 1 ,

lead respectively to

Sn+2=0, alln, (X)

Tn+1=0, alln, (Y)

whereSn+2denotes the determinant obtained fromAn+2by replacing its

last row by the row

[

cn+j− 1 cn+jcn+j+1···c 2 n+j

]

n+2

andTn+1denotes the determinant obtained fromBn+1by replacing its last

row by the row

[

cn+jcn+j+1cn+j+2···c 2 n+j

]

n+1

Regarding (X) and (Y) as conditions, what is their significance?

5.6 Distinct Matrices with Nondistinct Determinants

5.6.1 Introduction

Two matrices [aij]mand [bij]nare equal if and only ifm=nandaij=

bij,1≤i, j≤n. No such restriction applies to determinants. Consider