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# A Powerful Matrix

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Age 14 to 18

Challenge Level

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.*

This problem in geometry has been solved in no less than EIGHT ways by a pair of students. How would you solve it? How many of their solutions can you follow? How are they the same or different? Which do you like best?

Follow hints using a little coordinate geometry, plane geometry and trig to see how matrices are used to work on transformations of the plane.

Follow hints to investigate the matrix which gives a reflection of the plane in the line y=tanx. Show that the combination of two reflections in intersecting lines is a rotation.