Chapter III
Linear
equations and
inequalities
1-1
Many problems in economics can be modelled as a
system of linear equations or a system of linear
inequalities. In this chapter, we consider some basic
properties of such systems and discuss general solution
procedures.
1-2
Section I
SYSTEMS OF LINEAR
EQUATIONS
1-3
Preliminaries
At several points in previous chapters we have
been confronted with systems of linear
equations. For instance, deciding whether a set
of given vectors is linearly dependent or linearly
independent can be answered via the solution of
such a system. The following example of
determining feasible production programmes
also leads to a system of linear equations.
1-4
Example
•
1-5
Thus, we get the following system of three linear equations with four
variables:
1-6
•
1-7
Definition
The system (8.1) can also be written in matrix representation
Ax = b,
1-8
are the column vectors of matrix A.
1-9
1-10
Existence and uniqueness of a
solution
We investigate in which cases a system
has a solution. To this end, we introduce
the notion of the rank of a matrix.
THEOREM 8.1 Let A be a matrix of order
m × n. Then the maximum number of
linearly independent column vectors of A
coincides with the maximum number of
linearly independent row vectors of A.
1-11
r(A) = p ≤ min{m, n}.
THEOREM 8.2 The rank r(A) of matrix A is equal to the
order of the largest minor of A that is different from zero.
1-12
Example 1
1-13
Example 2
1-14
1-15
Augmented matrix
1-16
1-17
1-18
1-19
Elementary transformation;
solution procedures
Solution procedures for systems of linear
equations transform the given system into a
‘system with easier structure’. The following
theorem characterizes some
transformations of a given system of linear
equations such that the set of solutions
does not change.
1-20
Operations (1) to (3) are called elementary or equivalent
transformations
1-21
Basic solution
1-22
•
1-23
If matrix A is of order p ×n with r(A) = p < n, then at least
n−p variables are equal to zero in a basic solution of the
system Ax = b. The number of possible basic solutions of
a given system of linear equations is determined by the
number of different possibilities of choosing p basic
variables.
The methods typically used transform the original system
into either
(1) a canonical form according to Definition 8.7 ( pivoting
procedure or Gauss–Jordan elimination) or into
(2) a ‘triangular’ or echelon form (Gaussian elimination).
1-24
Pivoting procedure
we discuss the pivoting procedure. The
transformation of the original system into a
canonical form (possibly including less than m
equations) is based on the following theorem and
the remark given below.
1-25
Notice also that the first p rows in representation (8.2) describe a
system of linear equations in canonical form.
1-26
•
1-27
1-28
In particular, in this case we have the unique solution
1-29
•
1-30
We can rewrite the system in canonical
form in terms of the basic variables as
follows:
1-31
•
1-32
Example
Applying the pivoting procedure, we get the following
sequence of tableaus
1-33
1-34
From rows 13 to 16, we can rewrite the
system in terms of the basic variables:
1-35
Transformation formulas
1-36
1-37
Gaussian elimination
This procedure is based on the following
theorem
1-38
matrix A∗ (and matrix A too) does not have a minor of order
p+1 which is different from zero. This determinant is equal to
zero.
1-39
•
1-40
1-41
•
1-42
1-43
Example
Consider the following system of linear
equations:
Applying Gaussian elimination we obtain the
following tableaus.
1-44
1-45
1-46
Example
We solve the following system of linear
equations
Applying Gaussian elimination, we obtain the
following tableaus.
1-47
From row 9 we see that the considered system has no
solution since this equation
leads to a contradiction
1-48
Matrix inversion
The inverse X = A−1 of a matrix A of order
n × n satisfies the matrix equation AX = I ,
1-49
1-50
1-51
Example
We consider matrix
and determine the inverse A−1 by means of the pivoting
procedure
The computations are shown in the following scheme.
1-52
1-53
we obtain
1-54
Section 2
SYSTEMS OF LINEAR
INEQUALITIES
1-55
Example of system of
linear inequality
Which can be written, in a linear form as follows
1-56
1-57
1-58
Properties of feasible solutions
1-59
1-60
when considering a system of m inequalities with two
non-negative variables, the feasible region is described by the
intersection of m half-planes
1-61
THEOREM 8.10 The feasible region M of system (8.13) is
either empty or a convex set with at most a finite number of
extreme points.
1-62
Example
Two goods G1 and G2 are produced by
means of two raw materials R1 and
R2 with the capacities of 50 and 80 units,
respectively. To produce 1 unit of G1, 1
unit of R1 and 1 unit of R2 are required. To
produce 1 unit of G2, 1 unit of R1 and 2
units of R2 are required. The price of G1 is
3 EUR per unit, the price of G2 is 2 EUR
per unit and at least 60 EUR worth of
goods need to be sold.
1-63
Let xi be the number of produced units of Gi, i ∈ {1, 2}. A
feasible production programme has to satisfy the following
constraints:
This is a system of linear inequalities with only two variables,
which can be easily solved graphically.
The convex set of feasible solutions has five extreme points
described by the vectors xi (or points Pi), i = 1, 2, . . . , 5:
1-64
1-65
1-66
THEOREM 8.13 Any extreme point of the feasible region M of
system (8.13) corresponds to at least one basic feasible solution,
and conversely, any basic feasible solution corresponds
exactly to one extreme point.
1-67
A solution procedure
According to Theorem 8.11 we have to generate all
extreme points in order to describe the feasible region
of a system of linear inequalities. Using Theorems 8.13
and 8.14, respectively,
this can be done by generating all basic feasible
solutions of the given system.
In this section, we restrict ourselves to the case when
system (8.13) is given in the special form
Ax ≤ b, x ≥ 0, with b ≥ 0,
where A is an m×n matrix and we assume that r(A) =
m
1-68
Inequalities are transformed into equations by introducing a
slack variable in each constraint, i.e. for the ith constraint. We
assume that the set of feasible solution is a convex and
bounded.
where ui ≥ 0 is a so-called slack variable (i ∈ {1, 2, . . . ,m}).
as an initial basic feasible solution,
i.e. the variables x1, x2, . . . , xn are
the non-basic variables and the
slack variables u1, u2, . . . , um are
the basic variables
1-69
Example
Consider the system of linear inequalities presented in Example
8.10. Introducing slack variables u1, u2 and u3, we obtain the
initial tableau in rows 1 to 3 below. Now, the goal is to generate
all basic feasible solutions of the given system of linear
equations
1-70
1-71
Therefore, in the above example, there are six basic feasible
solutions and four basic infeasible solutions. From rows 1 to 18,
we get the following basic feasible solutions. (The basic
variables are printed in bold face.)
1-72
Deleting now the introduced slack variables u1, u2,
u3, we get the corresponding extreme points P1 with
the coordinates (0, 0), P2 with the coordinates (4, 0),
P3 with the coordinates (17, 13) and P4 with the
coordinates (10, 20). The fifth and sixth basic feasible
solutions correspond to extreme point P1 again. (In
each of them, exactly one basic variable has value
zero.) Therefore, the first, fifth and sixth basic
feasible solutions are degenerate solutions
corresponding to the same extreme point.
1-73
It can be seen that our computations have started from point P1,
then we have moved to the adjacent extreme point P2, then to
the adjacent extreme point P3 and finally to P4.
1-74