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) =
explanation: The document is a final exam for a linear systems course (MATH 152) from Spring 2021. It contains multiple math problems, each with subparts that require detailed solutions. The problems cover topics such as transition matrices, eigenvalues and eigenvectors, and systems of differential equations. Each problem is structured as a question with multiple sub-questions, but no answers are provided.
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distilabel_metadata.role: system
distilabel_metadata.content: # 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) =
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a final exam for a linear systems course (MATH 152) from Spring 2021. It contains multiple math problems, each with subparts that require detailed solutions. The problems cover topics such as transition matrices, eigenvalues and eigenvectors, and systems of differential equations. Each problem is structured as a question with multiple sub-questions, but no answers are provided.
  
  Label: multiple-question