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Consider Matrix $$ A = \begin{pmatrix} 1 & 1\\ 1 & 0 \end{pmatrix} $$

Investigate the sequence of powers of $A$ (i.e. $A^n$ for $n = 1, 2, 3, 4,\ldots$.

Verify that $$A^n = \begin{pmatrix}F_{n+1} &F_n \\ F_n & F_{n−1}\end{pmatrix}$$ for $n \geq 20$, where $F_n$ is the $n^{th}$ Fibonacci number.

I don't get it, please help. Thank you!

Jimin
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    How about trying to prove this statement by induction on $n$? – Amitesh Datta Oct 19 '15 at 12:47
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    What don't you get? You don't get what a matrix is? you don't get what $A^n$ means? you don't get what a Fibonacci number is? Meet us halfway – someone will help you. – Gerry Myerson Oct 19 '15 at 12:50
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    See also http://math.stackexchange.com/questions/61997/proof-of-this-result-related-to-fibonacci-numbers and http://math.stackexchange.com/questions/784710/how-to-prove-fibonacci-sequence-with-matrices – Martin Sleziak Oct 19 '15 at 13:43

2 Answers2

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Hint: $$A^{n+1} = A\cdot A^n = \begin{pmatrix} 1 & 1\\ 1 & 0 \end{pmatrix}\cdot\begin{pmatrix} F_{n+1} & F_{n}\\ F_{n} & F_{n-1} \end{pmatrix}$$$$$$ $$ = \begin{pmatrix} F_{n+1} + F_n & F_{n} + F_{n-1}\\ F_{n} + F_{n-1} & F_{n} \end{pmatrix} = \begin{pmatrix} F_{n+2} & F_{n+1}\\ F_{n+1} & F_{n} \end{pmatrix}$$

jameselmore
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Induction reasoning for $n\ge2$

$$\begin{bmatrix}1&1\\1&0\end{bmatrix}\times \begin{bmatrix}F_{n+1}&F_{n}\\F_{n}&F_{n-1}\end{bmatrix}=\begin{bmatrix}F_{n+2}&F_{n+1}\\F_{n+1}&F_{n}\end{bmatrix}$$

$$A^{2}=\begin{bmatrix}F_{2}&F_{1}\\F_{1}&F_{0}\end{bmatrix}$$

Arashium
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