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## 'Nine Eigen' printed from http://nrich.maths.org/

This problem involves the action
of matrices on vectors in three dimensions. The first few questions
look at fixed vector directions; the latter questions look at fixed
vectors. As you consider each point, make use of geometric or
algebraic arguments as appropriate. Draw diagrams and construct
particular examples of matrices and vectors if needed. If there is
no definitive answer to a given part, try to give examples of when
the question posed is or is not true.

In the questions below: $R, S$ are rotation matrices; $P, Q$ are
reflection matrices; $M$ is neither a rotation nor a
reflection.

- Which of the different types of matrices can leave no vector
directions fixed?
- Which of the different types of matrices can leave exactly one
vector direction fixed?
- Which of the different types of matrices can leave more than
one vector direction fixed?
- Is it ever the case that $RS$ can leave a vector
invariant?
- Is it ever the case that $PQ$ can leave a vector
invariant?
- Is it ever the case that $M$ will leave the direction of a
vector invariant?
- Can a matrix with determinant zero leave a vector fixed?
- Can a matrix with determinant greater than $1$ leave a vector
fixed?
- Can a matrix leave exactly two vectors fixed?