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IB Mathematics AI HL - Popular Quizzes

Eigenvalues, Eigenvectors & Matrix Powers

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Question 1

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hard

[Maximum mark: 6]

A discrete dynamical system is described by the following transition matrix, T\bm{T},

cf10a27b504c9599c9adc42101e9bc3c0f6527ac.svg

The state of the system is defined by the proportions of population with a particular characteristic.

  1. Use the characteristic polynomial of T\bm{T} to find its eigenvalues. [2]

  2. Find the corresponding eigenvectors of T\bm{T}. [2]

  3. Hence find the steady state matrix s\bm{s} of the system. [2]

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Question 2

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hard

[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 30%\text{\(30\)\hspace{0.05em}\%} of the mammals move from region X to region Y and 1515 % of the mammals move from region Y to region X. Assume that there are no mammal movements to or from any other neighboring regions.

  1. Write down a transition matrix T\bm{T} representing the movements between the two regions in a particular year. [2]

    1. Find the eigenvalues of T\bm{T}.

    2. Find a corresponding eigenvector for each eigenvalue of T\bm{T}.

    3. Hence write down matrices P\bm{P} and D\bm{D} such that T=PDP−1\bm{T} = \bm{PDP}^{-1}. [6]

Initially region X had 1260012\hspace{0.15em}600 and region Y had 1620016\hspace{0.15em}200 of these mammals.

  1. Find an expression for the number of mammals living in region Y after
    nn years, where n∈Z+n \in \mathbb{Z}^+. [5]

  2. Hence write down the long-term number of mammals living in region Y. [1]

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Question 3

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hard

[Maximum mark: 17]

Two grocery stores, store A and store B, serve in a small city. Each year, store A keeps 3030 % of its customers while 7070 % of them switch to store B. Store B keeps 60%\text{\(60\)\hspace{0.05em}\%} of its customers while 4040 % of them switch to store A.

  1. Write down a transition matrix T\bm{T} representing the proportions of the customers\text{customers} moving between the two stores. [2]

At the end of 20192019, store A had 83608360 customers while store B had 68206820 customers.

  1. Find the distribution of the customers between the two stores after two years.[2]

    1. Show that the eigenvalues of T\bm{T} are λ1=1\lambda_1 = 1 and λ2=−0.1\lambda_2 = -0.1.

    2. Find a corresponding eigenvector for each eigenvalue from part (c) (i).

    3. Hence express T\bm{T} in the form T=PDP−1\bm{T} = \bm{PDP}^{-1}. [6]

  2. Show that

    ba868786ed3787b5c4a0fd00e58f9ba6463b1270.svg

    , where n∈Z+n \in \mathbb{Z}^+. [2]

  3. Hence find an expression for the number of customers buying groceries from store A after nn years, where n∈Z+n \in \mathbb{Z}^+ [3]

  4. 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]

  5. Write down the long-term number of customers buying groceries from store A.[1]

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