The Chemistry Maths Book, Second Edition

17.2 Determinants of order 3 479

The minorM

21

is

Similarly,

The complete expansion of the determinant in terms of its elements is therefore

−a

21

1 M

21

1 + 1 a

22

1 M

22

1 − 1 a

33

1 M

33

1 = 1 −a

21

1 (a

12

a

33

1 − 1 a

13

a

32

)

• 1 a

22

1 (a

11

a

33

1 − 1 a

13

a

31

) 1 − 1 a

23

1 (a

11

a

32

1 − 1 a

12

a

31

)

and this is identical to the result (17.11) obtained by expansion along the first column.

EXAMPLE 17.4Find the value of the following determinant by expansion along

(a) the first row and (b) the third column:

(a) The expansion along the first row is

= 121 × 101 − 111 × 101 + 131 × 121 = 16

(b) The expansion along the third column is

0 Exercises 6–9

D=

−

−

3 −+=×=

42

11

00326

D=×

−

−×

−

+×

−

−

2

20

10

1

40

10

3

42

11

D=−

−

213

420

110

M

aa

aa

aa aa M

aa

22

11 13

31 33

11 33 13 31 23

11

==−,=

112

31 32

11 32 12 31

aa

=−aa aa

M

aa a

aa a

aa a

aa

aa

21

11 12 13

21 22 23

31 32 33

12

13

32 3

==

33

12 33 13 32

=−aa aa