Square pair

Explore the shape of a square after it is transformed by the action of a matrix.
Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative

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.