Chapter 7.2, Problem 27E

a. Based on your answers in Exercises 24 and 25, find closed formulas for the components of the dynamical system
$\stackrel{\to }{x}\left(t+1\right)=\left[\begin{array}{cc}0.5& 0.25\\ 0.5& 0.75\end{array}\right]\stackrel{\to }{x}\left(t\right)$ ,
with initial value ${\stackrel{\to }{x}}_0={\stackrel{\to }{e}}_1$ . Then do the same for the initial value ${\stackrel{\to }{x}}_0={\stackrel{\to }{e}}_2$ . Sketch the two trajectories.
b. Consider the matrix
$A=\left[\begin{array}{cc}0.5& 0.25\\ 0.5& 0.75\end{array}\right]$
Using technology, compute some powers of the matrix A, say, $A^2,A^5,A^{10},\mathrm{...}$ . What do you observe? Diagonalize matrix A to prove your conjecture. (Do not use Theorem 2.3.11, which we have not proven yet.)
c. If $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ is an arbitrary positive transition matrix, what can you say about the powers $A^t$ as t goes to infinity? Your result proves Theorem 2.3.11c for the special case of a positive transition matrix of size $2\times 2$ .