Prove that difference of next and previous terms of the Fibonacci sequence and the square of the current term is $\pm 1$

Question

Given the Fibonacci sequence $f_{n+1}=f_{n}+f_{n+1}$, how would I prove that \begin{equation} f_{n+1}*f_{n-1} - f_{n}^2 = \begin{cases} 1, & n \text{ odd}\\ -1, & n \text{ even} \end{cases} , \forall n \end{equation}

Answer 1

This is really easy if we notice that

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

with $F_0=0,F_1=1,F_2=1\dots$

Now notice that the determinant of the left matrix is $-1^n$ and the determinant of the matrix on the right is $F_{n+1}F_{n-1}-F_n^2$