<s>[INST] set problem of integers Consider the following set $F=\{F^0, F^1, F^2, \ldots\}$. This set consists of positive integers which satisfy the following properties:


*

*$F^0= F^1=1$ 
AND

*$F^n= F^{n-1} + F^{n-2}$ for all positive integers $n\geq2$.
Prove that for all positive integers $n$, the elements of the set $F$ satisfy the following identity:
$$\begin{vmatrix}
1 & 1 \\
1 & 0
\end{vmatrix}^{n+1}
=
\begin{vmatrix}
F^{n+1} & F^n \\
F^n & F^{n-1}
\end{vmatrix}$$
where $\begin{vmatrix}\end{vmatrix}$ denotes the determinant.
 [/INST] Hint:
$$\begin{pmatrix} F_{n+1}&F_n\\F_n&F_{n-1}\end{pmatrix}\begin{pmatrix} 1&1\\1&0
\end{pmatrix}=\begin{pmatrix} F_{n+1}+F_n&F_{n+1}\\F_{n}+F_{n-1}&F_n\end{pmatrix}
$$
</s>