Victor Camocho
math2250fall2011-2
WeBWorK assignment number Homework 13 is due : 12/01/2011 at 11:00pm MST.
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2
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1. (1 pt) Library/Rochester/setLinearAlgebra22SymmetricMatricesLet v1 = -3  , v2 = 2  , and v3 = 0 
/ur la 22 3.pg
3
0
1
be eigenvectors of the matrix A which correspond to the eigenFind the eigenvalues and associated unit eigenvectors of the
values
(symmetric)
matrix
 λ1 = −3, λ2 = −1, and λ3 = 2, respectively, and let
-4
-45 -15
A=
.
v = 1  .
-15 -5
smaller eigenvalue =
, -2
Express v as a linear combination of v1 , v2 , and v3 , and find Av.
,
associated unit eigenvector =
v= 
v1 + 
v2 +
v3 .
larger eigenvalue =
, .
Av =
.
associated unit eigenvector =
The above eigenvectors form an orthonormal eigenbasis for A.
2.
6.
(1 pt) Library/Rochester/setLinearAlgebra11Eigenvalues-
-2
-4
and v2 =
-3
3
are eigenvectors of a matrix A corresponding to the eigenvalues
λ1 = 2 and λ2 = 5, respectively,
/ur la 11 17a.pg
If v1 =
7 1
The matrix A =
-1 9
has one eigenvalue of multiplicity 2. Find this eigenvalue and
the dimenstion of the eigenspace.
eigenvalue =
,
dimension of the eigenspace =
.
Is the matrix A defective? (Type ”yes” or ”no”)
.
3.
then A(v1 + v2 ) =
(1 pt) Library/Rochester/setLinearAlgebra11Eigenvalues-
,
4.
.
For which value
of k does the matrix
-5 k
A=
9 -8
have one real eigenvalue of multiplicity 2?
k=
.

-1 0 2
The matrix A = 0 -3 0 
-2 0 -5
has one real eigenvalue. Find this eigenvalue and a basis of the
eigenspace.
eigenvalue
,
 =  

Basis:
and A(2v1 ) =
7.
(1 pt) Library/Rochester/setLinearAlgebra11Eigenvalues/ur la 11 14.pg
/ur la 11 18.pg

(1 pt) Library/Rochester/setLinearAlgebra11Eigenvalues-
/ur la 1126.pg
8.
(1 pt) Library/Rochester/setLinearAlgebra11Eigenvalues-
/ur la 11 2.pg
Findthe characteristic
 polynomial of the matrix
-1 -2 0
A = 0 1 -4  .
-4 1 0
p(x) =
.
.
(1 pt) Library/Rochester/setLinearAlgebra11Eigenvalues-
/ur la 11 27.pg
1
• D. If A is invertible, then A is diagonalizable.
9. (1 pt) Library/Rochester/setLinearAlgebra13ComplexEigenvalues14. (1 pt) Library/274/systems/prob84.pg
Solve the system dx
-2.5 1.5
=
x
-1.5 0.5
dt
3
with x(0) =
.
-1
Give your solution in real form.
x1 =
,
x2 =
.
/ur la 13 1.pg
The matrix -7 -7
A=
3 -8
has complex eigenvalues, λ1,2 = a ± bi, where a =
b=
.
10. (1 pt) Library/TCNJ/TCNJ Eigenvalues/problem7.pg
Determine if v is an eigenvector of the matrix A.
? 1. A =
? 2. A =
? 3. A =
-31
60
26
20
63
110
and
-18
-1
,v=
35 2
-30
0
,v=
-24 -3
-33
1
,v=
-58
2
[Note– you’ll probably want to view the phase plotter at
phase plotter (right click to open in a new window).
Select the ”integral curves utility” from the main menu. ]
If y0 = Ay is a differential equation, how would the solution
curves behave?
• A. All of the solution curves would run away from 0.
(Unstable node)
• B. The solution curves converge to different points.
• C. All of the solutions curves would converge towards
0. (Stable node)
• D. The solution curves would race towards zero and
then veer away towards infinity. (Saddle)
11. (1 pt) Library/TCNJ/TCNJ Diagonalization/problem1.pg
A, P and D are n × n matrices.
Check the true statements below:
• A. If there exists a basis for Rn consisting entirely of
eigenvectors of A, then A is diagonalizable.
• B. A is diagonalizable if A = PDP−1 for some diagonal
matrix D and some invertible matrix P.
• C. If A is diagonalizable, then A is invertible.
• D. A is diagonalizable if and only if A has n eigenvalues,
counting multiplicities.
15. (1 pt) Library/274/systems/prob66.pg
Solve the system dx
-0.5 -4
=
x
4 -0.5
dt
6
with x(0) =
.
8
Give your solution in real form.
x1 =
,
x2 =
.
12. (1 
pt) Library/TCNJ/TCNJ
Diagonalization/problem5.pg

2 0
0
Let: A = -9 7 12 
8 -6 -11
Find S, D and S−1 such that A = SDS−1 .
Starting
with the first row, put the eigenvalues
from
smallest
to



greatest.
S =
 D =


S−1 =

[Note– you’ll probably want to view the phase plotter at
phase plotter
(right click to open in a new window).
Select the ”integral curves utility” from the main menu. ]

? 1. Describe the trajectory.
? 1. What kind of interaction do we observe?
13. (1 pt) Library/TCNJ/TCNJ Diagonalization/problem2.pg
A, P and D are n × n matrices.
16.
(1
pt)
Library/Rochester/setDiffEQ13Systems1stOrder-
/ur de 13 6.pg
Match the differential equations and their vector valued function
solutions:
It will be good practice to multiply at least one solution out
fully, to make sure that you know how to do it, but you can get
the other answers quickly by process of elimination and just
multiply out one row element.
Check the true statements below:
• A. If A is diagonalizable, then A has n distinct eigenvalues.
• B. If AP = PD, with D diagonal, then the nonzero
columns of P must be eigenvectors of A.
• C. A is diagonalizable if A has n distinct eigenvectors.
2

1. y0 (t) =

2. y0 (t) =

3. y0 (t) =
-97
-140
-4
-13
-15
-33
-86
73
111
33
84
15
-2
-18
-18
218
-49
-138

-5
35  y(t)
-8

3
5  y(t)
7

-160
80  y(t)
165
A.

-2
y(t) = -4  e−21t
4
B.


-2
y(t) = 1  e45t
3
C.


-1
y(t) = 0  e−4t
-3
c
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Team, Department of Mathematics, University of Rochester
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