Copyright © University of Cambridge. All rights reserved.

## 'The Matrix' printed from http://nrich.maths.org/

One of the ways to work with transformations is to use a matrix. If
you have not met matrices before don't be put off, they are very
easy. In this question you will use some simple matrices for
rotations and reflections and see how they work.

First you need to know how to multiply a matrix like

$ \left( \begin{array}{cc} a &
b \\ c & d \end{array} \right) $ by the vector $ \left(
\begin{array}{c} x \\ y \end{array} \right) $

to give the image of the point $(x,y)$. This multiplication is
defined as follows: $$ \left( \begin{array}{cc} a & b \\ c
& d \end{array} \right) \left( \begin{array}{c} x \\ y
\end{array} \right) $$ $$ \left( \begin{array}{cc} a & b \\ c
& d \end{array} \right) \left( \begin{array}{c} x \\ y
\end{array} \right) = \left( \begin{array}{c} ax+by \\ cx+dy
\end{array} \right) $$ Find the images of the points $(1,0)$ and
$(0,1)$ under the transformation given by the matrix $$ \left(
\begin{array}{cc} a & b \\ c & d \end{array} \right) $$

Describe the effect on the plane of the four transformations where
$b=c=0$ and $a$ and $d$ take all possible combinations of the
values $\pm 1$. Now describe the effect on the plane of the four
transformations where $a=d=0$ and $b$ and $c$ take all possible
combinations of the values $\pm 1$

Explain why transformations have the same effect on the whole plane
as on the unit square with vertices $(0,0),\ (0,1),\ (1,1),\
(1,0)$.