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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