QUIZ 1 ANSWERS
1.
1
0
4
1
2

1
x
1
x
2
=
b
1
b
2
b
3
.
a).
(12 points) We reduce:
1
0
b
1
4
1
b
2
2

1
b
3
→
1
0
b
1
0
1
b
2

4
b
1
0

1
b
3

2
b
1
→
1
0
b
1
0
1
b
2

4
b
1
0
0
b
3
+
b
2

6
b
1
.
So the equation becomes:
1
0
0
1
0
0
x
1
x
2
=
b
1
b
2

4
b
1
b
3
+
b
2

6
b
1
.
b)
(6 points) Only when
b
3
+
b
2

6
b
1
=
0.
2. a)
(8 points) Solutions to
A
are length 3 column vectors, so
A
has three columns.
b)
(8 points) Any number. In fact consider
(
1

c

d
)
. This kills both of the given
vectors. Then, feel free to add any number of rows of zero’s below it.
c)
(8 points) We assume that the matrix is not zero as we are given that these are the
only special solutions. To find the rank, we note that each row of
A
must be in the subspace
of
R
3
which is orthogonal to
(
c
1
0
)
and
(
d
0
1
)
, but since these guys span a plane,
this subspace is a line. Thus the rows are all linearly dependant, and so the rank is one.
3. a)
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 Spring '11
 juan
 Linear Algebra, Matrices, Invertible matrix

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