[填空题]
An information technology (IT) m -j,br 3fjurc.f0i2qcompany offers pae (md6;p ntrpa(od409m 6s xkwid travelling vacation to its 160 employees every year. The employees can choose between travelling domestically or internationally. It is observed that 50 % of the employees who choose to travel domestically one year, choose internationally the next year. Conversely, 30 % of those who choose to travel internationally one year change to travel domestically the following year. For this year, 80 employees chose travelling domesticallyot mr9px w n6mds0;((p4dae6k and 80 employees chose travelling internationally.
1. Write down the initial state $\boldsymbol{s}_{0}$ and the transition matrix $\boldsymbol{T}$ .$\boldsymbol{T}=\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right]$ a = b = c = d = $\boldsymbol{s}_{0}=\left[\begin{array}{l}
a \\
b
\end{array}\right]$ a = b =
2. Determine $\boldsymbol{T} \boldsymbol{s}_{0}$ and interpret the result. $Ts_0$$\boldsymbol{s}_{0}=\left[\begin{array}{l}
a \\
b
\end{array}\right]$ a = b =
3. Find the eigenvalues and corresponding eigenvectors of $X_1$=$\boldsymbol{T}$ .$\boldsymbol{s}_{0}=\left[\begin{array}{l}
a \\
b
\end{array}\right]$ a = b = $X_2$=$\boldsymbol{T}$ .$\boldsymbol{s}_{0}=\left[\begin{array}{l}
a \\
b
\end{array}\right]$ a = b =
4. 1. Write an expression for the number of employees who choose travelling internationally after n years, n $\in \mathrm{N}$ .
2. Hence find the long term steady state number of employees to choose to travel internationally.
