Square Pair
Problem
You might like to look at Flipping Twisty Matrices before investigating this problem.
A 2D constant matrix $M$ acts on the plane, transforming each point with position vector ${\bf r}$ to another point with position vector $M{\bf r}$.
A particular square $S$ has corners with coordinates $(0,0), (0,1), (1,1), (1,0)$.
The action of the matrix $M$ on the points making up the square $S$ produces another shape in the plane.
What quadrilaterals can I transform $S$ into?
What quadrilaterals can I not transform $S$ into?
What shapes other than quadrilaterals can I transform $S$ into?
Another square $T$ has coordinates $(1,2), (2, 1), (3, 2), (2, 3)$.
When will $S$ and $T$ transform into the same type of quadrilateral? When will they transform into two different types of quadrilateral? Construct the matrices $M$ in each case.
You might like to use this Matrix Transformation tool to test out your ideas.
There are more matrix problems in this feature.
Getting Started
Student Solutions
Well done to Mateusz from Hautlieu School in Jersey, who sent in a full solution.
Mateusz began by investigating the shapes that $S$ can be transformed into:
Mateusz used co-ordinate geometry to investigate the possible shapes that are formed, but this could also be done using vectors.
Mateusz also considered some special cases which give more specific shapes:
Mateusz repeated the same steps for shape $T,$ and then found some rules for when $S$ and $T$ are transformed into the same shape:
There are more matrix problems in this feature.
Teachers' Resources
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 approaches leads to a better understanding of matrix algebra.
Possible approach
Students might like to use this Matrix Transformation tool to help them investigate the problem. In this tool the four corners of a quadrilateral are given as a $2 \times 4$ matrix, where the coordinates appear as the columns of the matrix, in clockwise (or anticlockwise) order.
Key questions
What can you say about the image of the points on a line 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.