The Chemistry Maths Book, Second Edition

484 Chapter 17Determinants

This result can be derived by a generalization of the method described in Section 17.2

for the system of three equations. Thus, forx

2

, multiplication of each equation of

(17.26) by the cofactor of the coefficient ofx

2

followed by addition givesDx

2

1 = 1 D

2

.

Then x

2

1 = 1 D

2

2 DifD 1 ≠ 10. We note that Cramer’s rule applies even when all the

quantitiesb

k

on the right sides of equations (17.26) are zero. In this case, all the

determinantsD

k

are also zero, and the solution is

x

1

1 = 1 x

2

1 = 1 x

3

1 =1-1= 1 x

n

1 = 10 (17.30)

Examples of the use of Cramer’s rule are Examples 17.1 and 17.2.

0 Exercises 15, 16

The case D 1 = 10

Cramer’s rule provides the unique solution of the system of linear equations (17.26)

so long as the determinant of the coefficients (17.27) is not zero. The rule does not

apply however whenD 1 = 10 because division byD 1 = 10 in (17.28) is not allowed. There

is then in general no unique solution or, in some cases, no solution of the equations at

all. For example, the system

(1) 2x 1 + 12 y 1 + z 1 = 110

(2) x 1 + 12 y 1 − 12 z 1 = 1 − 3 (17.31)

(3) 3x 1 + 12 y 1 + 14 z 1 = 120

has determinant

and no solution exists because the equations are inconsistent. Thus, addition of

equations (2) and (3) gives

4 x 1 + 14 y 1 + 12 z 1 = 117

whereas twice equation (1) is

4 x 1 + 14 y 1 + 12 z 1 = 120

The system (17.31) is made consistent by, for example, changing the right side of (3)

to 23:

(1) 2x 1 + 12 y 1 + z 1 = 110

(2) x+ 12 y 1 − 12 z 1 = 1 − 3 (17.32)

(3) 3x 1 + 12 y 1 + 14 z 1 = 123

D=−=

−

−

−

• =−−=

22 1

12 2

32 4

2

22

24

2

12

34

12

32

24 20 4 0