Anon

390 The Basics of financial economeTrics

formed by adding to the coefficient matrix a column formed with the con-
stant term—is represented as follows:

[]=

⋅⋅

⋅⋅⋅⋅⋅

⋅⋅

⋅⋅⋅⋅⋅

⋅⋅

⋅

⋅





      





      

aaa

aaa

aaa

b

b

b

Ab

jm

iijim

nnjnm

i

n

1,11,1,

,1 ,,

,1 ,,

1

If the constant terms on the right side of the equations are all zero, the
system is called homogeneous. If at least one of the constant terms is differ-
ent from zero, the system is said to be nonhomogeneous. A system is said to
be consistent if it admits a solution, that is, if there is a set of values of the
variables that simultaneously satisfy all the equations. A system is referred to
as inconsistent if there is no set of numbers that satisfy the system equations.
Let’s first consider the case of nonhomogeneous linear systems. The fun-
damental theorems of linear systems are listed as follows:

Theorem 1. A system of n linear equations in m unknown is consis-

tent (i.e., it admits a solution) if and only if the coefficient matrix

and the augmented matrix have the same rank.

Theorem 2. If a consistent system of n equations in m variables is

of rank r < m, it is possible to choose n − r unknowns so that the

coefficient matrix of the remaining r unknowns is of rank r. When

these m − r variables are assigned any arbitrary value, the value of

the remaining variables is uniquely determined.

The immediate consequences of the two fundamental theorems are that
(1) a system of n equations in n unknown variables admits a solution and
(2) the solution is unique if and only if both the coefficient matrix and the
augmented matrix are of rank n.
Let’s now examine homogeneous systems. The coefficient matrix and
the augmented matrix of a homogeneous system always have the same rank
and thus a homogeneous system is always consistent. In fact, the trivial solu-
tion x 1 =... = xm = 0 always satisfies a homogeneous system.
Consider now a homogeneous system of n equations in n unknowns. If
the rank of the coefficient matrix is n, the system has only the trivial solu-
tion. If the rank of the coefficient matrix is r < n, then Theorem 2 ensures
that the system has a solution other than the trivial solution.