= Solution
The initial vector decomposes especially simply in the <eigenbasis>:
$$
\begin{pmatrix}1\\2\end{pmatrix}=v_\varphi+v_\psi,
$$
because $\varphi+\psi=1$. Applying $A^n$ multiplies the two <eigenvectors> by $\varphi^n$ and $\psi^n$. Part (b) then yields
$$
\begin{pmatrix}L_{n+1}\\L_n\end{pmatrix}=\varphi^n\begin{pmatrix}\varphi\\1\end{pmatrix}+\psi^n\begin{pmatrix}\psi\\1\end{pmatrix}.
$$
Reading the second coordinate gives the <Lucas closed form>
$$
\boxed{L_n=\varphi^n+\psi^n=\left(\frac{1+\sqrt5}{2}\right)^n+\left(\frac{1-\sqrt5}{2}\right)^n\quad(n\geq0).}
$$
At $n=0$ this is 2 and at $n=1$ it is 1. Each summand obeys the same <recurrence relation>, which independently verifies the formula and explains why the irrational parts cancel to give an <integer>.
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