[填空题]
A biologist conducts an experiment u9qa uxw/f3 s-to study the pollinatiok/c axr /6y95yel7xftn preference of bumblebees' on different floral types. In a flight cage, 240 bumblebefly9xyk6ca r 7x5//et es are free to choose between two species of floral: A. majus striatum or A. majus pseudomajus. The changes of pollination behaviors of these bumblebees after every minute are reflected in the following table.
Initially, 150 bumblebees choose A . majus striatum and 90 bumblebees choose A . majus pseudomajus.
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 = $s_{0}=\left[\begin{array}{c}
a \\
b
\end{array}\right]$ a = b =
2. Determine $\boldsymbol{T} \boldsymbol{s}_{0}$ and interpret the result. $Ts_0$ = $=\left[\begin{array}{l}
a \\
b
\end{array}\right]$ a = b =
3. Find the eigenvalues and corresponding eigenvectors of $\boldsymbol{T}$ . $X_1$ $=\left[\begin{array}{l}
a \\
b
\end{array}\right]$ a = b = $X_2$$=\left[\begin{array}{l}
a \\
b
\end{array}\right]$ a = b =
4. 1. Write an expression for the number of bumblebees choosing to pollinate on A . majus pseudomajus after n minutes, n $\in \mathrm{N}$ .
2. Hence find the number of bumblebees choose to pollinate on A . majus pseudomajus in the long term.
