The closed form of the Fibonacci Sequence can be derived by
using matrices and computing the eigenvalues. That's because we
have the formula
$\left[\begin{matrix}F_n\\F_{n+1}\end{matrix}\right]=\left[\begin{matrix}0&1\\1&1\end{matrix}\right]\left[\begin{matrix}F_{n1}\\F_n\end{matrix}\right]$
That means
$\left[\begin{matrix}F_n\\F_{n+1}\end{matrix}\right]=\left[\begin{matrix}0&1\\1&1\end{matrix}\right]^n\left[\begin{matrix}0\\1\end{matrix}\right]$
and so if only we could compute powers rapidly and easily, then we could compute
the Fibonacci Sequence. But that's what eigenvectors and eigenvalues do for us.
