MATH 51 MIDTERM 2
February 26, 2004
1.
Find the inverse of the matrix
A
=
1

1
1

1
1
1
1
1

1
.
2.
Suppose
A
and
B
are 3x3 matrices, and that det(
A
) = 5 and det(
B
) =

2.
(a)
Find det(
AB
).
(b)
Find det(
A

1
).
(c)
Find det(2
A
).
3.
Let
A
=
•
0

1
2
3
‚
.
(a)
Find the eigenvalues of
A
.
(b)
Find the eigenvalues of
A
10
.
(c)
The matrix
A
=
2
1
1
1
3

2
1

2
3
has the number 3 as one of its eigenvalues. Find an eigenvector
v
that has 3 as its
associated eigenvalue.
4.
Let
T
:
R
2
→
R
2
be the linear transformation deﬁned by:
T
•
x
y
‚
=
•
x
+
y

2
x
+ 4
y
‚
.
(a).
Find the matrix
A
that represents the linear transformation
T
with respect
to the standard basis
S
=
{
e
1
,
e
2
}
.
(b).
Consider the basis
B
=
{
v
1
,
v
2
}
given by:
v
1
=
•
1
2
‚
,
v
2
=
•
3
7
‚
.
Find the change of basis matrix
C
for the basis
B
. That is, ﬁnd the matrix
C
such
that
v
=
C
[
v
]
B
for all vectors
v
.
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 '07
 Staff
 Linear Algebra, Algebra, Differential Calculus, Matrices, linear transformation

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