Determinants and Their Applications in Mathematical Physics

5.6 Distinct Matrices with Nondistinct Determinants 215

|α 2 |=−

∣ ∣ ∣ ∣ ∣ ∣

12 x

1 xx

2

φ 0 φ 1 φ 2

∣ ∣ ∣ ∣ ∣ ∣

=−

∣ ∣ ∣ ∣ ∣ ∣

1 x

1 φ 0 φ 1

xφ 1 φ 2

∣ ∣ ∣ ∣ ∣ ∣

(symmetric); (5.6.9)

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

|φ 3 |=−

∣ ∣ ∣ ∣ ∣ ∣ ∣

1 − 2 x

1 −xx

2

1 α 0 α 1 α 2

−xα 1 α 2 α 3

∣ ∣ ∣ ∣ ∣ ∣ ∣

=−

∣ ∣ ∣ ∣ ∣ ∣ ∣

1 − 3 x

1 − 2 x 3 x

2

1 −xx

2

−x

3

α 0 α 1 α 2 α 3

∣ ∣ ∣ ∣ ∣ ∣ ∣

,

|α 3 |=−

∣ ∣ ∣ ∣ ∣ ∣ ∣

12 x

1 xx

2

1 φ 0 φ 1 φ 2

xφ 1 φ 2 φ 3

∣ ∣ ∣ ∣ ∣ ∣ ∣

=−

∣ ∣ ∣ ∣ ∣ ∣ ∣

13 x

12 x 3 x

2

1 xx

2

x

3

φ 0 φ 1 φ 2 φ 3

∣ ∣ ∣ ∣ ∣ ∣ ∣

; (5.6.10)

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

∣

∣

∣

∣

φ 1 φ 2

φ 2 φ 3

∣

∣

∣

∣

=

∣ ∣ ∣ ∣ ∣ ∣

1 −xx

2

α 0 α 1 α 2

α 1 α 2 α 3

∣ ∣ ∣ ∣ ∣ ∣

,

∣

∣

∣

∣

α 1 α 2

α 2 α 3

∣

∣

∣

∣

=

∣ ∣ ∣ ∣ ∣ ∣

1 xx

2

φ 0 φ 1 φ 2

φ 1 φ 2 φ 3

∣ ∣ ∣ ∣ ∣ ∣

; (5.6.11)

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

∣

∣

∣

∣

φ 2 φ 3

φ 3 φ 4

∣

∣

∣

∣

=−

∣ ∣ ∣ ∣ ∣ ∣ ∣

1 − 2 x 3 x

2

1 −xx

2

−x

3

α 0 α 1 α 2 α 3

α 1 α 2 α 3 α 4

∣ ∣ ∣ ∣ ∣ ∣ ∣

=

∣ ∣ ∣ ∣ ∣ ∣ ∣

1 −xx

2

1 α 0 α 1 α 2

−xα 1 α 2 α 3

x

2

α 2 α 3 α 4

∣ ∣ ∣ ∣ ∣ ∣ ∣

,

∣

∣

∣

∣

α 2 α 3

α 3 α 4

∣

∣

∣

∣

=−

∣ ∣ ∣ ∣ ∣ ∣ ∣

12 x 3 x

2

1 xx

2

x

3

φ 0 φ 1 φ 2 φ 3

φ 1 φ 2 φ 3 φ 4

∣ ∣ ∣ ∣ ∣ ∣ ∣

=

∣ ∣ ∣ ∣ ∣ ∣ ∣

1 xx

2

1 φ 0 φ 1 φ 2

xφ 1 φ 2 φ 3

x

2

φ 2 φ 3 φ 4

∣ ∣ ∣ ∣ ∣ ∣ ∣

(5.6.12)

The coaxial nature of the determinantsB

nr

s

is illustrated for the case

(n, r)=(6,6) as follows:

∣

∣

∣

∣

φ 4 φ 5

φ 5 φ 6

∣

∣

∣

∣

=











each of the three determinants of order 6

enclosed within overlapping dotted frames

in the following display: