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# Flipping Twisty Matrices

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Age 14 to 18

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One of the ways to work with transformations is to use a matrix. If you have not met matrices before you might like to start by looking at the problem The Matrix.

Here is a reminder of how matrix multiplication works for a $2 \times 2$ matrix and a $2$D vector:

$$ \begin{pmatrix}a&b\\c&d\end{pmatrix} \begin{pmatrix}x\\y\end{pmatrix} =\begin{pmatrix}ax+by\\cx+dy\end{pmatrix} $$

If the matrix $\begin{pmatrix}a&b\\c&d\end{pmatrix}$ represents a transformation in the $x,y$ plane, then we can find the image of a point with coordinates $(p, q)$ by multiplying the transformation matrix and the position vector of the original point:

$$\begin{pmatrix}a&b\\c&d\end{pmatrix} \begin{pmatrix}p\\q\end{pmatrix}$$

**Example: Consider the transformation represented by the matrix $\begin{pmatrix}1&2\\0&1\end{pmatrix}$. By considering the image of the points $(0,0)$, $(1, 0)$, $(0, 1)$ and $(1, 1)$ describe what this transformation does to the unit square.**

The four points have images:

$\begin{pmatrix}1&2\\0&1\end{pmatrix}\begin{pmatrix}0\\0\end{pmatrix}=\begin{pmatrix}0\\0 \end{pmatrix}$

$\begin{pmatrix}1&2\\0&1\end{pmatrix}\begin{pmatrix}1\\0\end{pmatrix}=\begin{pmatrix}1\\0\end{pmatrix}$

$\begin{pmatrix}1&2\\0&1\end{pmatrix}\begin{pmatrix}0\\1\end{pmatrix}=\begin{pmatrix}2\\1\end{pmatrix}$

$\begin{pmatrix}1&2\\0&1\end{pmatrix}\begin{pmatrix}1\\1\end{pmatrix}=\begin{pmatrix}3\\1\end{pmatrix}$

So the unit square is transformed into a parallelogram (this type of transformation is called a*shear*).

You can find the image of the square in a single matrix multiplication as shown below:

$\begin{pmatrix}1&2\\0&1\end{pmatrix}\begin{pmatrix}0 & 0 & 1 & 1\\0 & 1 & 1 & 0\end{pmatrix}=\begin{pmatrix}0& 2& 3&1\\0 & 1 &1&0 \end{pmatrix}$

In this case I have listed the four vertices of the unit square in clockwise order, starting at $(0,0)$.

$\begin{pmatrix}1&2\\0&1\end{pmatrix}\begin{pmatrix}0\\0\end{pmatrix}=\begin{pmatrix}0\\0 \end{pmatrix}$

$\begin{pmatrix}1&2\\0&1\end{pmatrix}\begin{pmatrix}1\\0\end{pmatrix}=\begin{pmatrix}1\\0\end{pmatrix}$

$\begin{pmatrix}1&2\\0&1\end{pmatrix}\begin{pmatrix}0\\1\end{pmatrix}=\begin{pmatrix}2\\1\end{pmatrix}$

$\begin{pmatrix}1&2\\0&1\end{pmatrix}\begin{pmatrix}1\\1\end{pmatrix}=\begin{pmatrix}3\\1\end{pmatrix}$

So the unit square is transformed into a parallelogram (this type of transformation is called a

You can find the image of the square in a single matrix multiplication as shown below:

$\begin{pmatrix}1&2\\0&1\end{pmatrix}\begin{pmatrix}0 & 0 & 1 & 1\\0 & 1 & 1 & 0\end{pmatrix}=\begin{pmatrix}0& 2& 3&1\\0 & 1 &1&0 \end{pmatrix}$

In this case I have listed the four vertices of the unit square in clockwise order, starting at $(0,0)$.

Now consider matrices of the form

$$\begin{pmatrix}a&0\\0&d\end{pmatrix} $$

where $a$ and $d$ each take either the value $1$ or the value $-1$ (so there are four different matrices).

Explore what these four different transformations do. It would be a good idea to try different shapes, such as a square and a trapezium, for example the one with coordinates $(1,0)$, $(3, 0)$, $(2, 1)$ and $(0,1)$, so that you can tell whether a shape is reflected or rotated.

What happens if we consider instead matrices of the form $\begin{pmatrix}0&b\\c&0\end{pmatrix} $ where $b$ and $c$ can take the values $1$ and $-1$?

What if any of $a$, $b$, $c$ and $d$ can be equal to $1$, $-1$ or $0$?

What happens to the areas of the shapes under the different transformations?

*You might like to use this Matrix Transformation tool to help test out your ideas.*

*There are more matrix problems in this feature.*

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.