Transformations using a Matrix (Edexcel A Level Further Maths: Core Pure): Revision Note

Transformations using a matrix

What is a transformation matrix?

• A transformation matrix is used to determine the coordinates of an image from the transformation of an object

  • reflections, rotations, enlargements and stretches

  • Commonly used transformation matrices include

• (In 2D) a multiplication by any 2x2 matrix could be considered a transformation (in the 2D plane)

  • This can be done similarly in higher dimensions

• An individual point in the plane can be represented as a position vector, $\left(\begin{array}{c}x\\ y\end{array}\right)$

  • Several points, that create a shape say, can be written as a position matrix $\left(\begin{array}{cccc}{x}_1& x_2& x_3& ...\\ y_1& y_2& y_3& ...\end{array}\right)$

• A matrix transformation will be of the form $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)$

  • where $\left(\begin{array}{c}x\\ y\end{array}\right)$ represents any point in the 2D plane

  •  $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ is a given matrix

How do I find the coordinates of an image under a transformation?

• The coordinates (x’, y’) - the image of the point (x, y) under the transformation with matrix $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ are given by

$\left(\begin{array}{c}x\text{'}\\ y\text{'}\end{array}\right)=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)$

• Similarly, for a position matrix

$\left(\begin{array}{cccc}x{\text{'}}_1& x{\text{'}}_2& x{\text{'}}_3& ...\\ y{\text{'}}_1& y{\text{'}}_2& y{\text{'}}_3& ...\end{array}\right)=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\left(\begin{array}{cccc}{x}_1& x_2& x_3& ...\\ y_1& y_2& y_3& ...\end{array}\right)$ 

• A calculator can be used for matrix multiplication

  • If matrices involved are small, it may be as quick to do this manually 

• STEP 1 Determine the transformation matrix (T) and the position matrix (P) The transformation matrix, if uncommon, will be given in the question The position matrix is determined from the coordinates involved, it is best to have the coordinates in order, to avoid confusion 

• STEP 2 Set up and perform the matrix multiplication required to determine the image position matrix, P’
  P’ = TP 

• STEP 3 Determine the coordinates of the image from the image position matrix, P’

How do I find the coordinates of the original point given the image under a transformation?

•  To ‘reverse’ a transformation we would need the inverse transformation matrix

  • i.e. T^-1

  • For a 2x2 matrix $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ the inverse is given by $\frac{1}{\det \mathit{T}}\left(\begin{array}{cc}d& -b\\ -c& a\end{array}\right)$

    • where $\det \mathit{T}=ad-bc$

  • A calculator can be used to work out inverse matrices

• You would rearrange $\left(\begin{array}{c}x\text{'}\\ y\text{'}\end{array}\right)=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)$

  • $\frac{1}{\det \mathit{T}}\left(\begin{array}{cc}d& -b\\ -c& a\end{array}\right)\left(\begin{array}{c}x\text{'}\\ y\text{'}\end{array}\right)=\left(\begin{array}{c}x\\ y\end{array}\right)$

Determinant of a transformation matrix

What does the determinant of a transformation matrix (A) represent?

•  The absolute value of the determinant of a transformation matrix is the area scale factor (2D) or volume scale factor (3D)

  • Area scale factor = |det A| if 2x2

  • Volume scale factor = |det A| if 3x3

• The area/volume of the image will be product of the area/volume of the object and |det A|

  • Area of image = |det A| × Area of object (if 2x2)

  • Volume of image = |det A| × Volume of object (if 3x3)

• Note the area will reduce if |det A| < 1

• If the determinant is negative then the orientation of the shape will be reversed

  • For example: the shape has been reflected

How do I solve problems involving the determinant of a transformation matrix?

• Problems may involve comparing areas of objects and images

  • This could be as a percentage, proportion, etc

• Missing value(s) from the transformation matrix (and elsewhere) can be deduced if the determinant of the transformation matrix is known

• Remember to use the absolute value of the determinant

  • This can lead to multiple answers to equations

  • Use your calculator to solve these