[填空题]
Two grocery stores, store A and store B, serve in a w8ow h2(ql zw-smo 4tzfp53 ;etn*fia 93fvsj/vall city. Each year, store A keeps 30 % of its customers while 70 % of them switch 9fpt afnjt v4 ;ef 3z/o*si5v3to store B. Store B keeps 60% of its customers while 40 % of them switch to store A.
1.Write down a transition matrix T representing the proportions of the customers moving between the two stores.
$ T = \begin{vmatrix} a & b \\ c & d \end{vmatrix} $ ; a= ,b= ,c= ,d= .
At the end of 2019, store A had 8360 customers while store B had 6820 customers.
2.Find the distribution of the customers between the two stores after two years.
Store A:
Store B:
3.1.Show that the eigenvalues of T are $λ_1$ =1 and $λ_2$ =−0.1.
3.2.Find a corresponding eigenvector for each eigenvalue from part (c) (i).
$ X_1 = \begin{vmatrix} a \\ b \end{vmatrix} $ ; a= ,b= .
$ X_2 = \begin{vmatrix} c \\ -d \end{vmatrix} $ ; c= ,d= .
3.2.Hence express T in the form T=PDP$^{−1}$.
4.show that
where n∈Z$^+$.
5.Hence find an expression for the number of customers buying groceries from store A after n years, where n∈Z$^+$
6.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).
7.Write down the long-term number of customers buying groceries from store A.
The number is .
