A powerful matrix

What happens when you find the powers of this matrix?

Problem

Consider matrix ${\bf Q}$, where: 

$${\bf Q} = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}$$

Find ${\bf Q}^2$, ${\bf Q}^3$, ${\bf Q}^4$ and ${\bf Q}^5$.  What do you notice about the elements in your matrices?  Can you explain why this happens?

 

There are more matrix problems in this feature.