Page 325 - 35Linear Algebra

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2. (a) The augmented matrix

1 0 −1 2 −1
 
1 1 1 −1 2
 
 
 
 0 −1 −2 3 −3 
5 2 −1 4 1
encodes the system of equations.
(b) Again, write out the row operations as an additional exercise.
The above augmented matrix is row equivalent to
   
1 0 −1 2 −1 1 0 −1 2 −1
0 1 2 −3 3 0 1 2 −3 3
   
   
 0 −1 −2  0 0 0 0
  ∼  
3 −3  0 
0 2 4 −6 6 0 0 0 0 0
which is in reduced row echelon form.
(c) Solution set is

       
−1 1 −2
 
 
 
 3   −2   3 
X =   + µ 1   + µ 2   : µ 1 , µ 2 ∈ R .
 0   1   0 
 
 
0 0 1
 
 
−1
3
 
(d) The vector X 0 =   is a particular solution and the vectors
0
 
0
   
1 −2
 −2   3 
Y 1 =   and Y 2 =  
1 0
   
0 1
are homogeneous solutions. They obey
MX = V , MY 1 = 0 = MY 2 .
where
   
1 0 −1 2 −1
1 1 1 −1 2
   
M =   and V =   .
0 −1 −2 3 −3
   
5 2 −1 4 1
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