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Yiran from Guanghua Qidi in China chose a shape and looked at its image under matrices of the form $ \left( \begin{array}{cc} 0 & b \\ c & 0 \end{array} \right) $ where $b$ and $c$ can take the values $1$ and $-1.$ This is Yiran's work (click on the image to open a larger version). Note that for $b=1, c=-1$ the rotation is actually clockwise,
not anticlockwise - as shown in Yiran's diagram.
Andrei from School No. 205, Bucharest, Romania considered matrices of the form $ \left( \begin{array}{cc} a & 0 \\ 0 & d \end{array} \right) $ where $a$ and $d$ can take the values $1$ and $-1.$ Andrei also looked at more general points under both families of matrices. This is Andrei's work:
1. $b = c = 0,\ a = d = -1$ $$ \left( \begin{array}{cc} -1 & 0 \\ 0 & -1 \end{array} \right) \left( \begin{array}{c} x \\ y \end{array} \right) = \left( \begin{array}{c} -x \\ -y \end{array} \right) .$$ This means each point (x, y) transforms into its symmetrical image with respect to the origin, i.e. into the point (-x, -y). This is a rotation of
180 degrees about the origin.
2. $b = c = 0,\ a = - 1,\ d = 1$ $$ \left( \begin{array}{cc} -1 & 0 \\ 0 & 1 \end{array} \right) \left( \begin{array}{c} x \\ y \end{array} \right) = \left( \begin{array}{c} -x \\ y \end{array} \right) .$$ Here, it is a reflection in the y-axis.
3. $b = c = 0,\ a = d = 1$ $$ \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) \left( \begin{array}{c} x \\ y \end{array} \right) = \left( \begin{array}{c} x \\ y \end{array} \right) .$$ In this situation each point transforms into itself.
4. $b = c = 0,\ a = 1,\ d = -1$ $$ \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right) \left( \begin{array}{c} x \\ y \end{array} \right) = \left( \begin{array}{c} x \\ -y \end{array} \right) .$$ This transformation leaves the abscissa unchanged and modifies the sign of the ordinate, being a reflection in the x-axis.
Now, I look at the next set of transformations.
5. $a = d = 0,\ b = c = 1$ $$ \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right) \left( \begin{array}{c} x \\ y \end{array} \right) = \left( \begin{array}{c} y \\ x \end{array} \right) .$$ In this transformation, the abscissa and the ordinate are interchanged, the transformation being a reflection in respect to the line $y=x$, the angle
bisector of the first quadrant.
6. $a = d = 0,\ b = 1,\ c = -1$ $$ \left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right) \left( \begin{array}{c} x \\ y \end{array} \right) = \left( \begin{array}{c} y \\ -x \end{array} \right) .$$
Andrei also concluded that this is a rotation of $90^\circ$ clockwise about the origin.
7. $a = d = 0,\ b = c = -1$ $$ \left( \begin{array}{cc} 0 & -1 \\ -1 & 0 \end{array} \right) \left( \begin{array}{c} x \\ y \end{array} \right) = \left( \begin{array}{c} -y \\ -x \end{array} \right) .$$ This corresponds to a reflection in the line $y=-x$.
8. $a = d = 0,\ b = -1,\ c = 1$ $$ \left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right) \left( \begin{array}{c} x \\ y \end{array} \right) = \left( \begin{array}{c} -y \\ x \end{array} \right) .$$ This corresponds to a rotation of the point by 90 degrees anti-clockwise about the origin.
This problem in geometry has been solved in no less than EIGHT ways by a pair of students. How would you solve it? How many of their solutions can you follow? How are they the same or different? Which do you like best?
Follow hints using a little coordinate geometry, plane geometry and trig to see how matrices are used to work on transformations of the plane.
Follow hints to investigate the matrix which gives a reflection of the plane in the line y=tanx. Show that the combination of two reflections in intersecting lines is a rotation.