Page 75 - Matrix Analysis & Applied Linear Algebra

68 Chapter 2 Rectangular Systems and Echelon Forms

Observe that the system is indeed consistent because the last column is nonbasic.
Solve the reduced system for the basic variables x 1 ,x 2 , and x 5 in terms of the
free variables x 3 and x 4 to obtain

x 1 =1 − x 3 − 2x 4 ,
x 2 =1 − x 3 ,
x 3 is “free,”
x 4 is “free,”
x 5 = −1.
The general solution to the nonhomogeneous system is

     1   −1   −2 
x 1 1 − x 3 − 2x 4
1 − x 3
 x 2     1   −1   0 






    =  0  + x 3  1  + x 4  0  .

x 3
x =  x 3  = 
     0   0   1 
x 4 x 4
−1 −1 0 0
x 5
The general solution of the associated homogeneous system is
     −1   −2 
x 1 −x 3 − 2x 4
−x 3
 x 2     −1   0 
       
x 3  = x 3  1  + x 4  0  .
x =  x 3  = 
     0   1 
x 4 x 4
0 0 0
x 5
You should verify for yourself that
1
 
 1 
 
p =  0 
0
 
−1
is indeed a particular solution to the nonhomogeneous system and that
−1 −2
   
 −1   0 
and h 4 =  0 
   
h 3 =  1 
0 1
   
0 0
are particular solutions to the associated homogeneous system.