MATH 152 – LINEAR SYSTEMS, SPRING 2021
FINAL EXAM
Problem B1. (6 pts.) A supermarket sells three types of salads: Arugula, Baby Kale,
and Spring Mix. Every week, 40% of the people who bought Arugula last week switch
to Baby Kale, 20% switch to Spring Mix, and 40% buy Arugula again. Similarly, 20% of
customers who bought Baby Kale last week switch to Arugula, 10% switch to Spring Mix
and the remaining 70% buy Baby Kale again. Those who bought Spring Mix last week,
20% switch to Arugula, 20% switch to Baby Kale, and 60% stay with Spring Mix.
(a) (2 pts.) Find the transition matrix of the corresponding random walk.
(b) (2 pts.) Find the probability that a Spring Mix buyer will buy Spring Mix again
after two weeks. (Be careful when multiplying decimals, 0.2 · 0.2 = 0.04.)
(c) (2 pts.) In the long run, what will the share of each type of salad be? Give the
answer as a probability vector.
Problem B2. (6 pts.) Let
2
3
0 0 6
6
7
A = 41 3
25 .
0 0 3
(a) (3 pts.) Find the 3 eigenvalues of A. Some of them may be repeated.
(b) (3 pts.) Find 3 linearly independent eigenvectors of A.
Problem B3. (6 pts.) A system of di↵erential equations dtd ~y (t) = A~y (t) has the matrix
"
#
3 1
A=
.
2 1
(a) (3 pts.) Find the eigenvalues and corresponding eigenvectors of A.
(b) (2 pts.) Find the general solution "
of #the di↵erential equation in the real form.
1
(c) (1 pts.) Assuming that ~y (0) =
, find ~y (t). Your answer should not involve
0
complex numbers.