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# Matrix Meaning

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Age 16 to 18

Challenge Level

*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. All of the matrices are 2D matrices.

- 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 $MN = NM$?

What if the matrices are 3D matrices?

*There are more matrix problems in this feature.*