Math 136
Assignment 4 Solutions
1.
Let
A
=
1
0

1
2
1

2
,
B
=
3
2

1
0
1
0
1
3
3
,
C
=
2
1

2
.
Determine the following products or state that they are undefined.
Solution:
a)
AB
=
2

1

4
4

1

8
.
b)
BA
is undefined since
B
has 3 columns but
A
has only 2 rows.
c)
AC
=
4
9
d)
B
T
C
=
4

1

8
e)
C
T
C
=
9 .
f)
BA
T
=
4
10
0
1

2

1
2.
If AB is a 2
×
4 matrix, then what size are the matrices
A
and
B
?
Solution: Since
AB
is 2
×
4,
A
must have 2 rows and
B
must have 4 columns and the
number of columns of
A
must equal the number of rows of
B
. Hence
A
is 2
×
n
and
B
is
n
×
4.
1
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2
3.
Determine which of the following sets are linearly independent.
a)
S
=
{
(1
,

2
,
1
,
1)
,
(2
,

3
,
3
,
4)
,
(3
,

6
,
4
,
5)
}
.
Solution: Consider
(0
,
0
,
0
,
0) =
c
1
(1
,

2
,
1
,
1) +
c
2
(2
,

3
,
3
,
4) +
c
3
(3
,

6
,
4
,
5)
= (
c
1
+ 2
c
2
+ 3
c
3
,

2
c
1

3
c
2

6
c
3
, c
1
+ 3
c
2
+ 4
c
3
, c
1
+ 4
c
2
+ 5
c
3
)
This is the homogeneous system of linear equations with coe
ffi
cient matrix
A
=
1
2
3

2

3

6
1
3
4
1
4
5
.
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 Winter '08
 All
 Math, Linear Algebra, Vector Space, Elementary algebra

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