[填空题]
A city has two major sewut9 ya,ptguy 1o92k7curity guard companies, +ybeychb+bm z. ( :v-xcompany A and company B. Each year, 15 % of:z.+yc m- x(bbyhb+e v customers using company A move to company B and 5 % of the customers using company B move to company 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 .
