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Solutions to Homework #5, MAT1341A
Question # 1:
(2 + 2 + 2 = 6 points) Write down the general solutions of the systems with the following
augmented matrices.
(a)
±
1
2
0
1
3
0
0
1
0
4
²
Solution:
This is already in RREF; the nonleading variables are
x
2
and
x
4
so we set
x
2
=
r
and
x
4
=
s
;
then the two rows give us the equations
x
1
+ 2
x
2
+
x
4
= 3 (which yields
x
1
= 3

2
r

s
) and
x
3
= 4. Thus
the general solution is
x
1
= 3

2
r

s, x
2
=
r, x
3
= 4
, x
4
=
s
for
r, s
∈
R
or, in vector form
3

2
r

s
r
4
s

r, s
∈
R
(b)
1
0
0
1
0
1
0
2
0
0
0
0
0
0
0
3
Solution:
This is not in RREF (because the zero row is not at the bottom, for instance); but in any case
the last row corresponds to a degenerate equation, meaning that this system is inconsistent. The general
solution is the empty set.
(c)
³
0
0
0
0
1
0
´
Solution:
This system is the equation in 5 variables:
x
5
= 0. There are no conditions or constraints on any
of
x
1
, x
2
, x
3
or
x
4
, so these are all free, and the general solution is
{
(
r, s, t, u,
0)

r, s, t, u
∈
R
}
.
Note that this is the same thing you get if you follow the algorithm: this matrix is in RREF, with a leading 1
in the 5th column. Thus there are nonleading variables in the ﬁrst 4 columns, and we set their corresponding
variables equal to parameters; the value of
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 Winter '10
 DAIGLE
 Matrices

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