[填空题]
An information technoll+x4ufp xe e7y 8 k26gaw.pzs1ogy (IT) company offers paid trf1g6u*du 0dpoc4 q (:blxt 7p d/0wyk()nfhrgavelling 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 o gpp0/f:u4 dl ry7ht*1 dndo)cuqk0 (w6xgf (bne 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 domestically 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.
