Eigenvalues, eigenvectors and diagonalization

Eigenvalues, eigenvectors and steady state

[Maximum mark: 9]A discrete dynamical system is described by the following transition matrix, $T$$T=\begin{bmatrix}0.8&0.7\\0.2&0.3\end{bmatrix}$The state of the system is defined by the proportions of a population with a particular characteristic.(a) Use the characteristic polynomial of $T$ to find its eigenvalues.[3](b) Find the corresponding eigenvectors of $T$.[4](c) Hence find the steady-state matrix $S$ of the system.[2]