Chapter 4.8, Problem 7E

In Exercises 7-14, let $x_k=A{x}_{k-1}$, $k=1,2,\mathrm{.......}$ for the given $A$ and $x_0$. Find an expression for $x_k$ by using Eq. (6), as in Example 3. With a calculator, compute $x_4$ and $x_{10}$ from the expression. Comment on ${\lim }_{k\to \infty }{x}_k$.

$A$ and $x_0$ in Exercise $1$.

In Exercises 1-6, consider the vector sequence $\left\{{\text{x}}_{\text{k}}\right\}$, where ${\text{x}}_{\text{k}}=A{\text{x}}_{\text{k}-\text{1}}$, $k=1,2,\mathrm{........}$. For the given starting vector ${\text{x}}_{\text{0}}$, calculate $x_1$, $x_2$, $x_3$ and $x_4$ by using direct multiplication, as in Example $1$.

$A=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$, $x_0=\left[\begin{array}{c}2\\ 4\end{array}\right]$