The Chemistry Maths Book, Second Edition

17.3 The general case 481

17.3 The general case

A determinant of order nis a property of a square array ofn

2

elements, written

(17.21)

The determinant is defined forn 1 = 11 by

D 1 = 1 |a

11

| 1 = 1 a

11

(17.22)

It is defined forn 1 ≥ 12 by the expansion along any rowi(i 1 = 1 1, 2, =, n)

(17.23)

or, equivalently, by the expansion along any columnj(j 1 = 1 1, 2, =, n)

(17.24)

The quantity

C

ij

1 = 1 (−1)

i+j

M

ij

(17.25)

is the cofactor of elementa

ij

, andM

ij

is the minor ofa

ij

, the determinant of ordern 1 − 11

obtained from Dby deletion of row iand column j.

EXAMPLE 17.6Find the value of

Expansion along the first row gives

D=

−

−

−−

−

• 
  

−

−

−

2 −

20 1

10 2

31 1

1

40 1

10 2

01 1

1

421

112

031

3

4420

110

031

−

−

D=

−

−

−

2113

4201

1102

0311

DaC aC aC aC

j j j j nj nj

i

n

ij ij

=+ ++=

=

∑

11 2 2

1



DaC aC aC aC

i i i i in in

j

n

ij ij

=+ ++ =

=

∑

11 2 2

1



D

aaa a

aaa a

aaa a

n

n

n

=

11 12 13 1

21 22 23

2

31 32 33

3







aaa a

n

nn

1 nn

23