Determine the matrix for every n,$\begin{pmatrix}1&1\\1&0\end{pmatrix}^n$.

Question

$\begin{pmatrix}1&1\\1&0\end{pmatrix}^n$

Is the a formula which give us the matrix for every n? I should make a proof with induction.

Answer 1

Hint:

$\begin{pmatrix}1&1\\1&0\end{pmatrix}^n = \begin{pmatrix}F_{n+1}&F_n\\F_n&F_{n-1}\end{pmatrix}$

Where $F_n$ denotes the $n$-th Fibonacci number.