### 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.

### Key questions

When you perform two transformations, does the order
matter?

M and N are neither reflections nor rotations - what other
types of transformation could they represent?

### Possible extension

Construct matrices in three dimensions which make each
statement true or not true.

### Possible support

Transformations for
10 offers a chance to think about transformations
effected by different matrices.