Math 204. Final Exam (practice, solutions)
December 12th, 2010
This is a list of practice problems for the material we covered after exam 3. You should also
practice problems from previous exams since the final is cumulative.
1. Find the eigenvalues and eigenvectors for the matrix
A
=
1
2
2
1
.
The characteristic polynomial is
p
(
t
) =
t
2

2
t

3 = (
t

3)(
t
+ 1). So the eigenvalues are 3
and

1.
For
t
=

1.
A
+
I
=
2
2
2
2
. So
~x
∈
ker(
A
+
I
) if and only if 2
x
1
+2
x
2
= 0. So the eigenspace
is the span of the vector
1

1
. Now
A

3
I
=

2
2
2

2
. And the eigenspace is spanned by
the vector
1
1
.
Note
:
A
is symmetric, the eigenvalues of
A
are real, and the eigenvectors are perpendicular.
2. Let
p
be a polynomial, let
A
∈
C
n
×
n
, let
λ
be an eigenvalue for the matrix
A
.
Prove the
following:
(a)
p
(
λ
) is an eigenvalue of
p
(
A
). Suppose that
A~x
=
λ~x
for some nonzero vector
~x
. First
we prove that
A
k
~x
=
λ
k
~x
for any natural number
k
. The proof is by induction. The
case
k
= 1 is the assumption. Now assume the result for
k
and consider
A
k
+1
(
~x
) =
A
k
(
A
(
~x
)) =
A
k
(
λ~x
) =
λA
k
(
~x
) =
λλ
k
~x
=
λ
k
+1
~x.
Now let
p
(
t
) =
a
0
+
a
1
t
+
· · ·
+
a
m
t
m
. We have,
p
(
A
)
~x
= (
a
0
I
+
a
1
A
+
· · ·
+
a
m
A
m
)
~x
=
a
0
~x
+
a
1
A
(
~x
) +
· · ·
+
A
m
~x
)
=
a
0
~x
+
a
1
λ~x
+
· · ·
+
a
m
λ
m
~x
= (
a
0
+
a
1
λ
+
· · ·
+
a
m
λ
m
)
~x
=
p
(
λ
)
~x
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 Fall '08
 STAPLES
 Math, Linear Algebra, Algebra, Eigenvalue, eigenvector and eigenspace, Orthogonal matrix, Ker

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