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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.