Question

I need to prove for k, that \[A^k=\left[\begin{matrix}a^{k-1}-b^{k-1} & a^{k}-b^{k} \\ a^{k}-b^{k} & a^{k+1}-b^{k+1}\end{matrix}\right],k>0\] for \[A=\left[\begin{matrix}0 & 1 \\ 1 & 1\end{matrix}\right]\], where \[a=\frac{1-\sqrt5}{2},b=\frac{1+\sqrt5}{2}\]

Answer 1

I am thinking about using induction. And I have showed that this case is true for \[k=1\] Then I need to assume that this is true for all k, and then show it for \[k=k+1\] Am I right? And how would I do that.

Answer 2

Or rather, I have to use induction.

Answer 3

Lol, where are the qualified helpers?

Answer 4

oh goody, matrices

Answer 5

What am I even looking at? 0_0

Answer 6

For those having trouble loading the question: (I am )
I need to prove for k, that \[A^k=\left[\begin{matrix}a^{k-1}-b^{k-1} & a^{k}-b^{k} \\ a^{k}-b^{k} & a^{k+1}-b^{k+1}\end{matrix}\right],k>0\] for \[A=\left[\begin{matrix}0 & 1 \\ 1 & 1\end{matrix}\right]\], where \[a=\frac{1-\sqrt5}{2},b=\frac{1+\sqrt5}{2}\]

Answer 7

1 min

Answer 8

induction will work eventually