Solution
= Solution
For $n=1$,
$$
F_2F_0-F_1^2=-1=(-1)^1.
$$
If $C_n=F_{n+1}F_{n-1}-F_n^2$, then the recurrence gives
$$
\begin{aligned}
C_{n+1}
&=F_{n+2}F_n-F_{n+1}^2\\
&=(F_{n+1}+F_n)F_n-F_{n+1}^2\\
&=F_n^2-F_{n+1}F_{n-1}=-C_n.
\end{aligned}
$$
Induction yields the <Fibonacci determinant identity> known as Cassini's identity:
$$
\boxed{F_{n+1}F_{n-1}-F_n^2=(-1)^n}.
$$