Matrix meaning
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.
- 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?
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Getting Started
When considering the matrices in two dimensions, you could consider the algebraic form of the matrices or what happens geometrically. In two dimensions all rotations are around the origin. When considering reflections it is helpful is you don't restrict yourself to reflections in the $x$ and $y$ axes. To show that something is sometimes true and sometimes not true it is enough to show that there is one case where the statement is true, and one case where it isn't! You can sketch a diagram of a particular example to help show when something might, or might not, be true.
When thinking about rotations in three dimensions it may help to take an object and try turning it around different axes in different orders (make sure it isn't completely symmetrical - a ball wouldn't be of much use!). What happens? How does this relate to what you've been asked to do?
In some cases you might be able to use your examples from the two dimensional cases in the three dimensional cases as well.
Teachers' Resources
Why do this problem?
This problem asks students to consider the geometrical properties of matrix transformations in order to gain a greater understanding of matrix algebra, in 2 and 3 dimensions.
Possible approach
The problem works well as a discussion activity. Students could work with a partner and consider each statement first in 2D and then in 3D. After allowing them some time to consider the statements, work with examples, and think about the geometrical interpretation of the situation, bring the class together to discuss their ideas.
Encourage justifications which use geometrical reasoning as well as those using algebra. If a statement is sometimes true, it is important for students to identify when it is true, and geometrically speaking, why there are situations where it is and isn't true.
Encourage the use of sketches to show counterexamples!
Key questions
Possible extension
Possible support
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