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## 'Square Pair' printed from http://nrich.maths.org/

### Why do this problem?

This problem builds students' understanding of matrix
transformations in two dimensions and encourages exploration which
will increase confidence at working with vectors and matrices.
Insight gained from geometrical approachs leads to a better
understanding of matrix algebra.

### Possible approach

Students could explore this problem by choosing a variety of
different matrices and plotting the points which $(0,0), (0,1),
(1,0)$ and $(1,1)$ are transformed to. Once they have built up a
picture of how the square $S$ is transformed, they could be
challenged to find matrices which would transform $S$ into a
particular type of quadrilateral, and then to justify why some
types of quadrilateral cannot be made by transforming $S$. Students
may start by justifying these conjectures using their geometrical
insights, but they should be encouraged to support this using
appropriate matrix algebra.

The second part of the problem asks students to investigate
transformations of a second square; this could be done in the same
way, first by trying some numerical examples by choosing suitable
matrices, then generalising from what they find and supporting
their generalisations algebraically.

### Key questions

What can you say about the image of the points on a line after
transformation by a matrix?

What can you say about the image of a pair of parallel lines
after transformation by a matrix?

### Possible extension

Transformations for
10 offers a variety of challenging questions about the effects
of matrices in two and three dimensions, with an emphasis on
thinking geometrically.

### Possible support

Begin with lots of examples of transforming the
points $(0,0), (0,1), (1,0), (1,1)$ by multiplying by
different matrices. Plot the resulting four points each time, and
share ideas about what is common to all the images.