Page 230 - Determinants and Their Applications in Mathematical Physics

5.6 Distinct Matrices with Nondistinct Determinants 215

1 2x 1 x

|α 2 | = − 1 x x 2 = − 1 φ 0 (symmetric); (5.6.9)

φ 1
φ 0 φ 1 φ 2 xφ 1 φ 2

(n, r)=(4, 3):
1 1

−2x −3x
1 −x x 1 −2x
2 2
3 ,
3x
1 α 0 α 1 α 2 = − 1 −x x 2 −x
|φ 3 | = −

−xα 1 α 2 α 3 α 0 α 1 α 2 α 3
1 1

2x 3x
1 x 1
2 2
x 2x 3x
1 φ 0 φ 1 φ 2 1 x x x
|α 3 | = − = − 2 3 ; (5.6.10)

xφ 1 φ 2 φ 3 φ 0 φ 1 φ 2 φ 3
(n, r)=(3, 3):

1 −x x
2
φ 1

= α 0 α 1
φ 2 ,
φ 2 φ 3
α 2
α 1 α 2 α 3

1 x x 2

α 1

= φ 0 φ 1
α 2 ; (5.6.11)
α 2 α 3
φ 2
φ 1 φ 2 φ 3

(n, r)=(4, 4):
1 −2x 1 −x
2 2
3x x
2
φ 2 −x α 0 α 1 α 2
1 −x x 3 1
φ 3 ,
= − =
φ 3 φ 4 α 0 α 1 α 2 α 3 −xα 1 α 2 α 3

x 2
α 1 α 2 α 3 α 4 α 2 α 3 α 4
1 1 x
2 2
2x 3x x
2
1 x x 3 1

α 2 x φ 0 φ 1 φ 2

α 3 (5.6.12)
= −
=
α 3 α 4 φ 0 φ 1 φ 2 φ 3 φ 1 φ 2 φ 3
x
x 2
φ 1 φ 2 φ 3 φ 4 φ 2 φ 3 φ 4
The coaxial nature of the determinants B nr is illustrated for the case
s
(n, r)=(6, 6) as follows:

each of the three determinants of order 6


φ 4

= enclosed within overlapping dotted frames
φ 5

φ 5 φ 6 
in the following display:
