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# Nine Eigen

A vector ${\bf v}$ is fixed by a matrix if

$$

M{\bf v} = {\bf v}

$$

The vector direction of ${\bf v}$ is fixed if there is some number $\lambda\neq 0$ such that

$$ M{\bf v} = \lambda{\bf v} $$

This resource introduces concepts which appear in the later Further Pure Mathematics A Level modules.

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A vector ${\bf v}$ is fixed by a matrix if

$$

M{\bf v} = {\bf v}

$$

The vector direction of ${\bf v}$ is fixed if there is some number $\lambda\neq 0$ such that

$$ M{\bf v} = \lambda{\bf v} $$

This resource introduces concepts which appear in the later Further Pure Mathematics A Level modules.

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.