Prove the given property of the Fibonacci numbers I found in one of the books I read a lot of interesting properties of fibonacci numbers and among others this one in particular:
For all $n \in \mathbb N$, $F_{n+1}  F_{n-1} - F_n^2 = (-1)^n$.
I tried to use induction but at some point on induction step I just loose the point. Any ideas?

If $M = \pmatrix{0 & 1\cr 1& 1\cr}$, then $M^n = \pmatrix{F_{n-1} & F_n\cr
F_n & F_{n+1}\cr}$ (easy to prove by induction).  The left side of your equation is $\det(M^n) = (\det(M))^n$.
