The Chemistry Maths Book, Second Edition

480 Chapter 17Determinants

The cofactorC

ij

of elementa

ij

is the minorM

ij

multiplied by the appropriate sign,

C

ij

1 = 1 (−1)

i+j

1 M

ij

(17.18)

The expansion (17.15) along the first column is then

More generally, the expansion along the jth column is

(17.19)

and along the ith row is

(17.20)

An expansion in terms of cofactors is called a Laplace expansionof the determinant

EXAMPLE 17.5Find the cofactors of the elements of the third column of

We have

0 Exercise 10

aM CM

33 33 33 33

0

21

42

== 88

−

=− =+ =−

aM CM

23 23 23 23

0

21

11

== 33

−

==−=−

aM CM

13 13 13 13

3

42

11

== 22

−

−

==+=

213

420

110

−

−

aaa

aaa

aaa

aC aC a

ii i i

11

12 13

21 22 23

31 32 33

11 2 2

=+ +

iii

j

ij ij

CaC

33

1

3

=

=

∑

aaa

aaa

aaa

aC aC a

jj j j

11

12 13

21 22 23

31 32 33

11 2 2

=+ +

333

1

3

jj

i

ij ij

CaC=

=

∑

aaa

aaa

aaa

aC aC a

11

12 13

21 22 23

31 32 33

11 11 21 21

=+ +

3 31 31

1

3

11

CaC

i

ii

=

=

∑