The Chemistry Maths Book, Second Edition

17.2 Determinants of order 3 477

and, expanding the second-order determinants,

D 1 = 1 a

11

a

22

a

33

1 − 1 a

11

a

23

a

32

1 − 1 a

21

a

12

a

33

1 + 1 a

21

a

13

a

32

1 + 1 a

31

a

12

a

23

1 − 1 a

31

a

13

a

22

(17.11)

The solution of the system of three equations can then be expressed in terms of

third-order determinants as

(17.12)

whereD 1 ≠ 10 is the determinant of the coefficients, (17.10), and

(17.13)

EXAMPLE 17.2Use determinants to solve the equations

2 x 1 − 13 y 1 + 14 z 1 = 18

y 1 − 13 z 1 = 1 − 7

x+ 12 y 1 + 12 z 1 = 111

The determinant of the coefficients is, by equation (17.10),

= 121 × 181 − 101 × 1 (−14) 1 + 111 × 151 = 121

The determinantsD

1

,D

2

,andD

3

are, by equations (17.13),

Thereforex 1 = 1 D

1

2 D 1 = 1 1,y 1 = 1 D

2

2 D 1 = 1 2,z 1 = 1 D

3

2 D 1 = 13.

0 Exercise 5

DD

12

834

71 3

11 2 2

21

284

073

111 2

= 42

−

−−=,=−−=, DD

3

238

017

1211

= 63

−

−=

D=

−

−=

−

−

−

• 
  

−

−

234

013

122

2

13

22

0

34

22

1

34

13

D

ba a

ba a

ba a

D

aba

ab

1

1

12 13

22223

33233

2

11

113

21

=,=

2223

31 3 33

3

11

12 1

21 22 2

31 32 3

a

aba

D

aab

aab

aab

,=

x

D

D

x

D

D

x

D

D

1

1

2

2

3

3

=, = , =