Physical Chemistry 3: — Chemical Kinetics — - Christian-Albrechts ...

Appendix D 280
D.5 Systems of linear equations
I Solutions of linear equations (I): Cramer’s rule. Matrices and determinants can
be generally used for solving systems of linear equations of the type
11 1 + 12 2 + 13 3 + 1 = 1
21 1 + 22 2 + 23 3 + 2 = 2
=
(D.39)
1 1 + 2 2 + 3 3 +
=
These equations can be written in matrix form:
⎛
⎞ ⎛ ⎞ ⎛ ⎞
11 12 1 1 1
⎜ 21 22 2
⎟ ⎜ 2
⎟
⎝ . . . ⎠ ⎝ . ⎠ =
⎜ 2
⎟
⎝ . ⎠
1 2
or
Ax= c
(D.40)
(D.41)
Simple algebraic manipulations (see any math textbook) gives the expressions
|A| = |A |
(D.42)
where |A| is the determinant of the coefficient matrix A and |A | is the respective
determinant of |A| in which the th column is replaced by the 1 2 3 ,e.g.
|A 2 | =
⎛
⎞
11 1 1
⎜ 21 2 2
⎟
⎝ . . . ⎠
1
(D.43)
Solutions for the : From Eq. D.42 we obtain the non-trivial solutions for the
according to
= |A |
(D.44)
|A|
under the condition that the A matrix is not singular, i.e., the determinant of coefficients
does not vanish
|A| 6= 0
(D.45)
(The trivial and uninteresting solutions are 1 2 3 =0).
I Solutions of linear equations (II): In the following, we shall only consider the special
linear equations of the type
11 1 + 12 2 + 13 3 + 1 =0
21 1 + 22 2 + 23 3 + 2 =0
= .
1 1 + 2 2 + 3 3 + =0
(D.46)