MAS 3105
June 18, 2009
Final Exam and Key
Prof. S. Hudson
1a) [each part of problem 1 is 5 points]. In the Rabbit example, we studied
L
:
R
2
→
R
2
where
L
([
a, b
]
T
) = [
a
+
b,
2
a
]
T
. Find the matrix representation
A
of
L
(w.r.t. the std basis
of
R
2
).
1b) We found two eigenvectors of
A
,
x
1
= [1
,
1]
T
and
x
2
= [1
,

2]
T
, which form a basis
X
of
R
2
. Find the corresponding eigenvalues.
1c) Find the matrix representation
B
of
L
w.r.t. the basis
X
.
1d) Find an eigenvector for
B
. Is it also an eigenvector of
A
? Explain briefly.
1e) Let
x
= [2
,
3]
T
X
(the coordinates are wrt the eigenvector basis,
X
).
Find [
L
(
x
]
S
, in
standard coordinates.
2) [20 pts] TrueFalse. You can assume the matrices are all square.
A
T
A
and
AA
T
always have the same rank.
If
A
and
B
are unitary then
AB
is unitary.
If
A
is singular then
AB
is singular.
If
A
H
=

A
then
A
is normal.
If
A
is unitary then
A
is not defective.
If
A
is Hermitian and unitary then
A
2
=
I
.
If
U
,
V
are subspaces of
R
n
, and
U
⊥
V
then
V
⊂
U
⊥
.
If
A
is unitary and
x
∈
C
n
then

A
x

=

x

.
If
λ
is an eigenvalue of a unitary matrix then

λ

= 1.
If
λ
is an eigenvalue of a unitary matrix then
λ
is real.
3) [10pts] Suppose
A
is a 4x4 matrix and det
A
= 3. Find det (adj(
A
)).
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 Spring '09
 JULIANEDWARDS
 Matrices, Orthogonal matrix, λ, 1 qt, Prof. S. Hudson

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