Square Pair
Age 16 to 18
Challenge Level
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
$(0,0), (0,1), (1,0), (1,1)$.
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 $(4,4), (6, 2), (8, 4), (6,
6)$.
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.