The Chemistry Maths Book, Second Edition

478 Chapter 17Determinants

Minors and cofactors

The minorM

ij

of elementa

ij

of a determinant Dis the determinant obtained by

deleting row iand column jof D. For example, (17.14) shows the result of deleting

row 2 and column 3 of a third-order determinant.

(17.14)

In general, the minors of a determinant of order nare determinants of ordern 1 − 11.

They are important because they are used for the expansion of a determinant in terms

of its elements. Thus, equation (17.10) can be written as

(17.15)

This is called expansion along the first column; each element of the first column is

multiplied by its minor, and the products are added with appropriate signs. The sign

associated with elementa

ij

is

(17.16)

The signs for the third-order determinant are

(17.17)

A determinant can be expanded along anyrow or column.

EXAMPLE 17.3Expand a determinant of order 3 along the second row.

The elements of the second row area

21

,a

22

anda

23

. Therefore, making use of (17.17)

for the signs,

row 2

11 12 13

21 22 23

31 32 33

21 21

→=−+

aaa

aaa

aaa

aM aaM aM

22 22 23 23

−

+−+

−+−

+−+

()−=

++

−+











+

1

1

1

ij

ij

ij

if is even

if is odd

aaa

aaa

aaa

aM aM a

11

12 13

21 22 23

31 32 33

11 11 21 21

=− +

3 31 31

M

aa a

aa a

aa a

aa

aa

M

11 12 13

21 22 23

31 32 33

11

12

31 32

2

==

33