The Chemistry Maths Book, Second Edition

17.4 The solution of linear equations 483

NOTE: on the evaluation of determinants and the solution of linear equations.

The expansion of a determinant in terms of its elements consists of n! products of

nelements at a time, and involves n!(n 1 − 1 1)multiplications (see (17.11) for n 1 = 13 ).

It follows therefore that such expansions do notprovide a practical (or accurate)

method for the evaluation of large determinants. The same is true of the use of

Cramer’s rule for the solution of linear equations discussed in the following section.

Except for very small values of n, numerical methods such as those described in

Section 17.6 and Chapter 20 must always be used.

17.4 The solution of linear equations

We have seen in Sections 17.1 and 17.2 that the solutions of two linear equations,

(17.6), and of three linear equations, (17.12), can be written as ratios of determinants

when the determinant of the coefficients of the unknowns is not zero. In general, the

system of nlinear equations,

a

11

x

1

1 + 1 a

12

x

2

1 + 1 a

13

x

3

1 + 1  1 + 1 a

1 n

x

n

1 = 1 b

1

a

21

x

1

1 + 1 a

22

x

2

1 + 1 a

23

x

3

1 + 1  1 + 1 a

2 n

x

n

1 = 1 b

2

a

31

x

1

1 + 1 a

32

x

2

1 + 1 a

33

x

3

1 + 1  1 + 1 a

3 n

x

n

1 = 1 b

3

(17.26)

a

n 1

x

1

1 + 1 a

n 2

x

2

1 + 1 a

n 3

x

3

1 + 1  1 + 1 a

nn

x

n

1 = 1 b

n

has a single unique solution if the determinant of the coefficients,

(17.27)

is not zero. This solution is given by Cramer’s rule,

3

(17.28)

whereD

k

is obtained from Dby replacement of the kth column of Dby the column

with elementsb

1

, b

2

, =, b

n

. For example,

(17.29)

x

D

DD

aba a

aba a

aba a

n

n

2

2

11

113

1

21 2 23 2

31 3 33

1

==







33

1

3

n

n

nn

nn

aba a



x

D

D

x

D

D

x

D

D

n

n

1

1

2

2

=, = ,... =

D

aaa a

aaa a

aaa a

n

n

n

=

11

12 13

1

21 22 23

2

31 32 33

3







aaa a

n 1 nn 23 nn

3

Gabriel Cramer (1704–1752), Swiss, published the rule in his Introduction à l’analyse des lignes courbes

algébriques(Introduction to the analysis of algebraic curves) in 1750.