Let $A=\left[\begin{array}{cc}1& 2\\ 2& 1\end{array}\right]$

a$)$ Find the two eigenvalues ${}_1$ and ${}_2$ and their respective eigenvectors vec$(x_1)$ and vec$(x_2).$

b$)$ Build a matrix $P$ such that $P=\left[\begin{array}{cc}\frac{?}{\text{b}\text{a}\text{r}}(x_1)& \text{v}\text{e}\text{c}(x_2)\end{array}\right]$ and find its inverse $P^{-1}.$

c$)$ Build a diagonal matrix $D$ such that $D=\left[\begin{array}{cc}{}_1& 0\\ 0& {}_2\end{array}\right]$ and find $D^n.$

d$),$ Find $A^n.)=(P{D}^n{P}^{-1}$