IB Mathematics AI HL - Popular Quizzes
Eigenvalues, Eigenvectors & Matrix Powers
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Question 1
[Maximum mark: 6]
A discrete dynamical system is described by the following transition matrix, ,
The state of the system is defined by the proportions of population with a particular characteristic.
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Use the characteristic polynomial of to find its eigenvalues. [2]
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Find the corresponding eigenvectors of . [2]
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Hence find the steady state matrix of the system. [2]
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Question 2
[Maximum mark: 14]
Zoologists have been collecting data about the migration habits of a particular species of mammals in two regions; region X and region Y. Each year of the mammals move from region X to region Y and % of the mammals move from region Y to region X. Assume that there are no mammal movements to or from any other neighboring regions.
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Write down a transition matrix representing the movements between the two regions in a particular year. [2]
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Find the eigenvalues of .
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Find a corresponding eigenvector for each eigenvalue of .
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Hence write down matrices and such that . [6]
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Initially region X had and region Y had of these mammals.
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Find an expression for the number of mammals living in region Y after
years, where . [5]
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Hence write down the long-term number of mammals living in region Y. [1]
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Question 3
[Maximum mark: 17]
Two grocery stores, store A and store B, serve in a small city. Each year, store A keeps % of its customers while % of them switch to store B. Store B keeps of its customers while % of them switch to store A.
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Write down a transition matrix representing the proportions of the moving between the two stores. [2]
At the end of , store A had customers while store B had customers.
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Find the distribution of the customers between the two stores after two years.[2]
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Show that the eigenvalues of are and .
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Find a corresponding eigenvector for each eigenvalue from part (c) (i).
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Hence express in the form . [6]
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Show that
, where . [2]
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Hence find an expression for the number of customers buying groceries from store A after years, where [3]
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Verify your formula by finding the number of customers buying groceries from store A after two years and comparing with the value found in part (b). [1]
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Write down the long-term number of customers buying groceries from store A.[1]
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