Question: If $A=\left[\begin{array}{ccc} 1 & 0 & 0\\ 1 & 0 & 1\\ 0 & 1 & 0\end{array}\right]$ then show that $A^n=A^{n-2}+A^2-I$, $n\geq3$. Hence find $A^{50}$.
Where do I begin? I solved upto $A^8=4A^2-I$ and $A^9=4A^2+A-I$ with similar pattern emerging but I'm stuck where to go from here. And since I'm starting from the equation I was supposed to arrive at, I don't think this is the solution. But I don't see anything I can do with $A$ to prove that equation except maybe calculate $A^2$...