text: # 2021 Final Written
  
  ```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 &middot; 0.2 = 0.04.)
  (c) (2 pts.) In the long run, what will the share of each type of salad be? Give the
  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
  &quot;
  #
  3 1
  A=
  .
  2 1
  (a) (3 pts.) Find the eigenvalues and corresponding eigenvectors of A.
  (b) (2 pts.) Find the general solution &quot;
  of #the di↵erential equation in the real form.
  1
  (c) (1 pts.) Assuming that ~y (0) =