## estudier Group Title Prove Cassini's identity using the matrix representation of the Fibonacci sequence one year ago one year ago

1. JamesWolf Group Title

oh is that all? easy! one question, whats cassini's identity? To google!

2. estudier Group Title

Heh, I can open questions all over the shop now.....:-)

3. JamesWolf Group Title

4. JamesWolf Group Title

out of interest. whats the formula for the fibonacci sequence

5. estudier Group Title

Where's the matrix?

6. JamesWolf Group Title

$F_n = F_{n-2} + F_{n-1}$ ?

7. JamesWolf Group Title

oh sorry yes your right there is none there

8. MrMoose Group Title

You can represent the Fibonacci sequence as: $\left[\begin{matrix}F_{n+1} & F_n \\ F_n &F_{n-1}\end{matrix}\right] = \left[\begin{matrix}1 & 1 \\ 1 & 0\end{matrix}\right]^n$

9. MrMoose Group Title

This is fairly easy to prove by induction

10. MrMoose Group Title

define F_0 as 0, F_1 as 1, and F_2 as 1

11. MrMoose Group Title

If the first equation is true, then$\left[\begin{matrix}1 & 1 \\ 1 & 0\end{matrix}\right]^{n+1}=\left[\begin{matrix}1 & 1 \\ 1 & 0\end{matrix}\right]^n*\left[\begin{matrix}1 & 1 \\ 1 & 0\end{matrix}\right]$: $= \left[\begin{matrix}F_{n+1} & F_n \\ F_n & F_{n-1}\end{matrix}\right]*\left[\begin{matrix}1 & 1 \\ 1 & 0\end{matrix}\right]$ $= \left[\begin{matrix}F_{n+1}+F_n & F_{n+1} \\ F_n+F_{n-1} & F_n\end{matrix}\right]=\left[\begin{matrix}F_{n+2} & F_{n+1} \\ F_{n+1} & F_n\end{matrix}\right]$

12. MrMoose Group Title

So: $\left[\begin{matrix}F_{n+1} & F_n \\ F_n & F_{n-1}\end{matrix}\right] = \left[\begin{matrix}1 & 1 \\ 1 & 0\end{matrix}\right]^n \forall n \in Z_+$

13. MrMoose Group Title

now, take the determinant of both sides: $F_{n+1}F_{n-1}-F_n^2=\det \left[\begin{matrix}1 & 1 \\ 1 & 0\end{matrix}\right]^n$

14. MrMoose Group Title

The determinant of the products of square matrices is the products of the determinants of the matrices: $= (\det \left[\begin{matrix}1 & 1 \\ 1 & 0\end{matrix}\right])^n$

15. MrMoose Group Title

=$(0-1)^n=-1^n$