本题目来源于试卷: IB MAI HL Statistics & Probability Topic 4.3 Probability，类别为 IB数学
[问答题]
Zoologists have been collecting data about thegm3m du6bvtiod 77 j..a- :/f ilo+xlr mif5 v bqbp6z +j9dys75ggration habits of a particular species of mammals in two regions; region X and region Y. Each year 30 %6d bzy9g+pqfs5jvb 57 of the mammals move from region X to region Y and 15 % 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 $\boldsymbol{T}$ representing the movements between the two regions in a particular year.
2. 1. Find the eigenvalues of $\boldsymbol{T}$ .
2. Find a corresponding eigenvector for each eigenvalue of $\boldsymbol{T}$ .
3. Hence write down matrices $\boldsymbol{P}$ and $\boldsymbol{D}$ such that $\boldsymbol{T}=\boldsymbol{P D} \boldsymbol{P}^{-1}$ .
Initially region $\mathrm{X}$ had 12600 and region $\mathrm{Y}$ had 16200 of these mammals.
3. Find an expression for the number of mammals living in region Y after n years, where n $\in \mathbb{Z}^{+}$ .
4. Hence write down the long-term number of mammals living in region Y .
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