B U Department of Mathematics
Math 201 Matrix Theory
Summer 2005 Final Exam
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1.(a)
[3] A 4
×
4 matrix C is known to have eigenvalues
λ
1
= 2
, λ
2
=
λ
3
=

3 and
λ
4
= 4.Find
det(
I
+
C
),
Trace
(
I
+
C
) and det(exp
C
)
Solution:
If C has eigenvalues 2,3,3,4 than (I+C) has eigenvalues 3,2,2,5 and exp(
C
) has eigenvalues
e
2
, e

3
, e

3
, e
4
then we have
det(
I
+
C
) = 3
.
(

2)
.
(

2)
.
5 = 60
Trace
(
I
+
C
) = 3 + (

2) + (

2) + 5 = 4
det(
e
C
) =
e
2
.e

3
.e

3
.e
4
= 1
(b)
[3] If K is a skewsymmetric square matrix show that
Q
= (
I

K
)(
I
+
K
)

1
is an orthogonal
matrix.
Solution:
Compute
Q
T
Q
.
Q
T
= [(
I
+
K
)

1
]
T
[
I

K
]
T
= (
I
+
K
T
)

1
(
I

K
T
) = (
I

K
)

1
(
I
+
K
)
Q
T
Q
= (
I

K
)

1
(
I
+
K
)(
I

K
)(
I
+
K
)

1
=
I
since (
I
+
K
)(
I

K
) = (
I

K
)(
I
+
K
) therefore Q is orthogonal.
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 Fall '08
 SOYSAL
 Math, Linear Algebra, Matrices, Eigenvalue, eigenvector and eigenspace, Orthogonal matrix, linearly independent vectors

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