Page 76 - Matrix Analysis & Applied Linear Algebra

2.5 Nonhomogeneous Systems 69

Now turn to the question, “When does a consistent system have a unique
solution?” It is known from (2.5.7) that the general solution of a consistent
m × n nonhomogeneous system [A|b] with rank (A)= r is given by

h n−r ,
x = p + x f 1 h 1 + x f 2 h 2 + ··· + x f n−r
where

x f 1 h 1 + x f 2 h 2 + ··· + x f n−r h n−r
is the general solution of the associated homogeneous system. Consequently, it
is evident that the nonhomogeneous system [A|b] will have a unique solution
(namely, p ) if and only if there are no free variables—i.e., if and only if r = n
(= number of unknowns)—this is equivalent to saying that the associated ho-
mogeneous system [A|0] has only the trivial solution.
Example 2.5.2

Consider the following nonhomogeneous system:

2x 1 +4x 2 +6x 3 =2,
x 1 +2x 2 +3x 3 =1,
x 1 + x 3 = −3,
=8.
2x 1 +4x 2
Reducing [A|b]to E [A|b] yields

246 2 100 −2
   
 123 1   010 3 
101 −3 001 −1
[A|b]=   −→   = E [A|b] .
240 8 000 0
The system is consistent because the last column is nonbasic. There are several
ways to see that the system has a unique solution. Notice that

rank (A) = 3 = number of unknowns,

which is the same as observing that there are no free variables. Furthermore,
the associated homogeneous system clearly has only the trivial solution. Finally,
because we completely reduced [A|b]to E [A|b] , it is obvious that there is only
 
−2
one solution possible and that it is given by p =  3  .
−1