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M346 Second Midterm Exam, October 23, 2003
1.
Find all the eigenvalues of the following matrices.
You do NOT need to
ﬁnd the corresponding eigenvectors.
[Note:
the answers are fairly simple,
and can be obtained without a lot of calculation, using the various “tricks of
the trade”.]
a)
3
1
5
17
1
3
4

10
0
0
2

1
0
0
1
2
b)
1
2
3
4
0
2
1
2
3
.
2. The eigenvalues and eigenvectors of the matrix
A
=
0

1

1

1
0

1

1

1
0
are
λ
1
=

2,
λ
2
= 1 and
λ
3
= 1, and corresponding eigenvectors
b
1
=
1
1
1
,
b
2
=
1

1
0
and
b
3
=
1
0

1
.
(That is, the eigenvalue 1 has
multiplicity two, and a basis for the eigenspace
E
1
is
{
b
2
,
b
3
}
.)
a) Solve the diﬀerence equation
x
(
n
+ 1)
=
A
x
(
n
) with initial condition
x
(0) =
3
0
0
(which equals
b
1
+
b
2
+
b
3
, by the way). That is, ﬁnd
x
(
n
) for
every
n
.
b) With the situation of part (a), identify the stable, unstable, and neutrally
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This note was uploaded on 04/10/2011 for the course M 346 taught by Professor Radin during the Spring '08 term at University of Texas at Austin.
 Spring '08
 RAdin
 Linear Algebra, Algebra, Eigenvectors, Vectors, Matrices

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