TITLE: Pattern with $\begin{bmatrix}0&1\\-1&3\end{bmatrix}^k$. How can I prove what corner element will be?
QUESTION [6 upvotes]: The matrix
$$B= \begin{bmatrix}0&1\\-1&3\end{bmatrix}$$
Is inspired by this matrix which generates the Fibonacci sequence.
Upon calculating a few powers of $B$, I noticed that $({B^k})_{2,2}$ seems to be ${F_{2(k+1)}} - {F_{2k}}$ but I lack an idea of how to prove it.
What would be a fruitful approach to do it?

REPLY [3 votes]: Hint: Prove by induction that
$$
\begin{bmatrix}\hphantom-0&1\\-1&3\end{bmatrix}^k
=
\begin{bmatrix}-F_{2k-2} & F_{2k} \\ -F_{2k} & F_{2k+2} \end{bmatrix}
$$
You'll need that $F_{n+2} = 3F_n-F_{n-2}$. For a proof, adapt this answer. See also this question.