Matrix meaning

Explore the meaning behind the algebra and geometry of matrices with these 10 individual problems.
Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative

Problem



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.

 

  1. Is it always the case that $M+N = N + M$?

     
  2. It it always the case that $RS= SR$?

     
  3. It it always the case that $RP= PR$?

     
  4. It it always the case that $PQ= QP$?

     
  5. Is it ever the case that $MN = NM$?

 

What if the matrices are 3D matrices?

There are more matrix problems in this feature.