Square Pair
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.
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