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Science, Technology, Engineering and Mathematics
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Explore the meaning behind the algebra and geometry of matrices with these 10 individual problems.
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.
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.
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?
Fix Me or Crush Me
investigates matrices which fix certain vectors and vectors which are fixed by certain matrices.
provides some simpler questions about the possible effects of different types of $3 \times 3$ matrices.
Maths Supporting SET
Addition of vectors
Eigenvalues and eigenvectors
Meet the team
The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
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