[填空题]
A city has two major security g f-ama;y 11n1kjnm q;)nfvq /puard companies, company A and companq+af*ae(uv o 0pn- (upy B. Each year, 15 % of customers using company A move to company B and 5 % of the customers using company B move to companyepunauoqf+ -0va ((*p A. All additional losses and gains of customers by the companies can be ignored.
1.Write down a transition matrix T representing the movements between the two companies in a particular year.
$ T = \begin{vmatrix} a & b \\ c & d \end{vmatrix} $ ; a= ,b= ,c= ,d= .
2.1.Find the eigenvalues and corresponding eigenvectors of T.
$ X_1 = \begin{vmatrix} a \\ b \end{vmatrix} $ , a= ,b= ;
$ X_2 = \begin{vmatrix} c \\ -d \end{vmatrix} $ , c= ,d= .
2.2.Hence write down matrices P and D such that T=PDP$^{-1}$.
Initially company A and company B both have 3600 customers.
3.Find an expression for the number of customers company A has after n years, where n∈Z.
Hence the number of customers company A has after n years is
C(n)=a+b(c0.8$^n$) ; a , b= , c= .
4.Hence write down the number of customers that company A can expect to have in the long term.
The number is .
