1 point to lose here!
9
Bates
11 12 Elkin
Name:
Student number:
M229 Matrices and Linear Equations
Third Examination
Version SAMPLE
Test date: Monday November 17
Testing: Weber 134 9 AM - 4PM AND Glover 130 6:30 PM - 9 PM – BRING your CSU ID
Y
(4 pts) 1. A linear function fA transforms the in−−→
dicated figure. The matrix A has OP
as an eigenvector with eigenvalue −1 and
−−→
OQ as an eigenvector with eigenvalue 1/2.
Sketch the transform plot.
Y
P
-1
X
-1
Q
(4 pts) 1. The matrix A =
1
1
4 −2
. has eigenvalues −3, 2. Compute its components A−3 and A2 .
(6 pts) 2. Does A have λ = −3 as an eigenvalue? If
so find all associated basic eigenvectors.


41 −16
12
12  .
A =  44 −19
−88
32 −27


−9 −8 −3
3. The matrix A =  17
17
7  has minimal polynomial µA (x) = (x + 1)2 (x + 2)
−26 −28 −12
(2 pts) A)
(2 pts) B)
(6 pts) C)
(2 pts) D)
Is the matrix diagonalizable?
What are its eigenvalues?
Find the basic eigenvectors associated with each eigenvalue.
Write the associated matrix P and diagonal matrix D such that AP = P D.
(over)
X
Version SAMPLE




−15
6
−6
4
6
−3  and v =  2 .
4. Let A =  −9
5
−18 21
−11
(6 pts) A) Compute the A-annihilator f (x) of v.
(2 pts) B) Is f (x) the minimal polynomial of A?
2
5. Let wn be the number of walks of length n from vertex
1 to vertex 4 in the indicated digraph.
(2 pts) A) Compute w1 , w2 , w3 , w4 by inspection.
(6 pts) B) Write the associated transfer matrix and compute a
recurence relation satisfied by {wn }.
(1 pt) C) Use this recurrence relation to find w5 .
5. A statistical study consisting of 4 samples of 2 attributes is summarized by the data matrix:
1
4
3
M=
0 1
1 1
7 0
3 3
.
(4 pts) A) Compute the standardized data matrix.
(5 pts) B) Compute the correlation matrix and use the calculator to estimate its eigenvalues.
6. Serena studies squirrels. She observes that squirrels 0-1 years old produce 0.2 offspring on average and 50%
survive 1 year. Squirrel 1-2 years old produce an average of 1.2 offspring and 80% survive 2 years. Squirrels
over 2 years produce an average of .6 offspring. This model assumes that squirrels do not live more than 3
years.
(5 pts) A) Find the transition matrix A for this population model.
(5 pts) B) Compute the spectral radius of A and describe what this model predicts about the squirrel popuation
over the long term.
c
2008
Department of Mathematics Colorado State Univeristy