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For each section of questions, ask students to think about what they are being asked to do, use their intuition to make any initial comments, then think about the geometry of the situation and finally use some examples to support their thoughts algebraically.
Nine Eigen
Age 16 to 18
Challenge Level 
Why do this problem?
This problem asks a series of questions designed to provoke students' thinking about matrices which leave vectors fixed, and the properties that such matrices and vectors would have.Possible approach
It may be worthwhile to start with some preliminary work about
matrices in three dimensions. Students could find some examples of
$3 \times 3$ matrices which represent simple rotations and
reflections, which could be used in answering the problem.
The questions divide neatly into three sections - questions
1-3, 4-6 and 7-9. Students could tackle these questions in those
three sections, perhaps working with a partner, and feed back ideas
to the rest of the class after each section is answered.
For each section of questions, ask students to think about what they are being asked to do, use their intuition to make any initial comments, then think about the geometry of the situation and finally use some examples to support their thoughts algebraically.
Key questions
What can you say about a rotation that leaves the direction of
a vector unchanged?
What can you say about a reflection that leaves the direction
of a vector unchanged?
Possible extension
Fix Me or
Crush Me investigates matrices which fix certain
vectors and vectors which are fixed by certain
matrices.
Possible support
Matrix Meaning provides some simpler questions about the possible effects of different types of $3 \times 3$ matrices.You may also like
8 Methods for Three by One
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Rots and Refs
Follow hints using a little coordinate geometry, plane geometry and trig to see how matrices are used to work on transformations of the plane.
Reflect Again
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.