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Explore how matrices can fix vectors and vector directions.
This problem involves the algebra of matrices and various geometric concepts associated with vectors and matrices. As you consider each point, make use of geometric or algebraic arguments as appropriate. 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 five questions below: $R, S$ are rotation matrices; $P, Q$ are reflection matrices; $M,N$ are neither rotations nor reflections. Consider each part in 2D and 3D.
Is it always the case that $M+N = N + M$?
It it always the case that $RS= SR$?
It it always the case that $RP= PR$?
It it always the case that $PQ= QP$?
Is it ever the case that $MN = NM$?
How do the values of the determinants of the various matrices affect the results of these questions?
Eigenvalues and eigenvectors
Maths Supporting SET
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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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