Science:Math Exam Resources/Courses/MATH152/April 2022/Question B2 (c)
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Question B2 (c) 

This is a continuation of B2 (a). (c) In the long run, what will the share of each type of apple be? 
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Hint 

If the share of each apple stabilizes in the long run, how should the transition matrix act on the vector of shares? 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Suppose that, in the long run, are the proportions of buyers who buy Ambrosia, Braeburn and Cameo apples, respectively. Then the transition matrix should leave the vector fixed. In other words, we are looking for an eigenvector with eigenvalue 1, or an element of the nullspace of the matrix We can find the elements of the nullspace by rowreducing. Again, we indicate the row operations as in B1 (a), but we also use for the operation "swap rows and ". We can easily see from the rowreduced matrix above that eigenvectors of with eigenvalue 1 are of the form If we want the entries of this vector to be proportions of people buying the three different kinds of apples, then the entries of the eigenvector must sum up to 1. The equation implies that , so we have that the longrun shares of each apple variety are 