<s>[INST] Fibonacci and Matrices 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! 
 [/INST] 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}$$
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