Determine the matrix for every n,$\begin{pmatrix}1&1\\1&0\end{pmatrix}^n$.

Question

$\begin{pmatrix}1&1\\1&0\end{pmatrix}^n$

Is the a formula which give us the matrix for every n? I should make a proof with induction.

What is the problem? How we can find a value for $n$ without some condition? — Emilio Novati, Sep 29 '15 at 21:20
I have to find the matrix for every n. It has to be a formula. — Zauberkerl, Sep 29 '15 at 21:23
How can I find the n-th matrix without n matrix multiplication? — Zauberkerl, Sep 29 '15 at 21:26
Why don't you do like, 3 matrix multiplications, and see if a pattern pops up? — pjs36, Sep 29 '15 at 21:56
I see the pattern, but I can't make a proof for the whole matrix. I should probably make a mathematical induction — Zauberkerl, Sep 29 '15 at 21:57
See also http://math.stackexchange.com/questions/61997/proof-of-this-result-related-to-fibonacci-numbers-beginpmatrix11-10-end and http://math.stackexchange.com/questions/784710/how-to-prove-fibonacci-sequence-with-matrices — Martin Sleziak, Oct 19 '15 at 14:22

score 3 · Accepted Answer · answered Sep 29 '15 at 21:22

3

Hint:

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

Where $F_n$ denotes the $n$-th Fibonacci number.

answered Sep 29 '15 at 21:22

Victor

4

Is this really a hint or a solution? – David Quinn Sep 29 '15 at 22:08
2

I see it as a hint more than a solution. My intention was to show the OP where to look for in order to start an inductive argument. – Victor Sep 29 '15 at 22:10

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