$(1$ point$)$ Note: In this problem, use the method from class. See the lecture notes for January $12-17$ on Brightspace. Also see Example $8\mathrm{.}54$ in the textbook. More examples $($using real matrices$)$ are in Section $8\mathrm{.}6$ of the textbook.

Consider the sequence defined recursively by

$F_0=0,F_1=3,F_{n+2}=2F_{n+1}-5F_n.$

We can use matrix diagonalization to find an explicit formula for $F_n.$

$($a$)$ Find a matrix A that satisfies

$\left[\begin{array}{c}{F}_{n+1}\\ F_{n+2}\end{array}\right]=A\left[\begin{array}{c}{F}_n\\ F_{n+1}\end{array}\right]$

$A=[]$

$($b$)$ Find the appropriate exponent $k$ such that

$\left[\begin{array}{c}{F}_n\\ F_{n+1}\end{array}\right]=A^k\left[\begin{array}{c}{F}_0\\ F_1\end{array}\right]$

$k=$

$($c$)$ Find a diagonal matrix $D$ and an invertible matrix $P$ such that

$A=PD{P}^{-1}.$

$($d$)$ Find an explicit closed formula for Fn