Determinants and Their Applications in Mathematical Physics

218 5. Further Determinant Theory

Once again the symbol←→ denotes that the columns are arranged in

reverse order.

Illustrations

B 2 (x)=

[

1

1 x

]

,

B 3 (x)=





1

12 x

13 x 4 x

2



,

B 4 (x)=







1

13 x

15 x 9 x

2

16 x 19 x

2

27 x

3





,

B 5 (x)=











1

14 x

17 x 16 x

2

19 x 37 x

2

64 x

3

110 x 55 x

2

175 x

3

256 x

4











.

Sincebijis a function ofn,Bnis not a submatrix ofBn+1. Finally, define

a block matrixN 2 nof order 2nas follows:

N 2 n=

[

O

Sn(x)

Bn(x) An

]

, (5.6.23)

whereAn=[αm]n, as before.

Illustrations

N 4 =







1

1 x

1 α 0 α 1

1 xα 1 α 2





,

N 6 =

      

1

13 x

1 xx

2

1 α 0 α 1 α 2

12 xα 1 α 2 α 3

13 x 4 x

2

α 2 α 3 α 4

      

N 2 nis symmetric only whenn=2.

A subset ofN 4 is: