[填空题]
Zoologists have been c7f)r: s-v f28.skrzr5x at4rd h+wockollectitd/p)87jze5d,jyneao uy7 rh)xil 20ng data about the migration habits of a particular species of mammals in two regions; region X and region Y. Each year 30% 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 ot)pej,dr27 xayzh 5je87/unyld)0 oither neighboring regions.
1.Write down a transition matrix T representing the movements between the two regions in a particular year.
$ T = \begin{vmatrix} a & b \\ c & d \end{vmatrix} $ ; a= , b= , c= , d= .
2.1.Find the eigenvalues of T.
Hence the eigenvalues of T are λ$_1$= and λ$_2$= .
2.2.Find a corresponding eigenvector for each eigenvalue of T.
$ X_1 = \begin{vmatrix} a \\ b \end{vmatrix} $ ; a= , b= .
$ X_2 = \begin{vmatrix} A \\ -B \end{vmatrix} $ ; A= , B= .
2.3.Hence write down matrices P and D such that T=PDP$^{-1}$.
$ P = \begin{vmatrix} a & b \\ c & -d \end{vmatrix} $ ; a= , b= , c= , d= .
$ D = \begin{vmatrix} A & B \\ C & D \end{vmatrix} $ ; A= , B= , C= , D= .
Initially region X had 12600 and region Y had 16200 of these mammals.
3.Find an expression for the number of mammals living in region Y aftern years, where n∈Z$^+$ .
4.Hence write down the long-term number of mammals living in region Y.
The number is .
