Enlargement Matrices (Edexcel IGCSE Maths)

Spec point — 5G

Enlargement matrices

How do I find enlargement matrices?

• Imagine the unit square OABC

  • It has a side-length of 1 unit
  • O is the origin

• The coordinates of A and C as column vectors are

  • $A = ( \begin{gathered} 1 \\ 0 \end{gathered} )$ and $C = ( \begin{gathered} 0 \\ 1 \end{gathered} )$

• Under an enlargement of scale factor k with centre at the origin(including negative scale factors), A moves to A' and C moves to C'

  • The matrix, M representing this enlargement is $M = ( \begin{gathered} A ' \vert & C ' \end{gathered} )$

  • A' and C' are column vectors of the new positions

    • So $M$ is a 2×2 matrix

  • The points O and B are not needed, as we can draw the enlarged square using just A' and C' (as O won't move)

• $A ' = ( \begin{gathered} k \\ 0 \end{gathered} )$ and $C ' = ( \begin{gathered} 0 \\ k \end{gathered} )$

  • They are both just moving along the x and y axes respectively

• So all enlargement matrices have the form $M = ( \begin{gathered} k 0 \\ 0 k \end{gathered} )$

  • This is the same as $M=kI$, where $I$ is the identity matrix

• For example:

  • The matrix representation of an enlargement of scale factor 3 with centre at the origin is $( \begin{gathered} 3 0 \\ 0 3 \end{gathered} )$
  • The matrix representation of an enlargement of scale factor $-\frac{1}{2}$ with centre at the origin is $( \begin{gathered} - 1 \frac{1}{2} & 0 \\ 0 - 1 \frac{1}{2} \end{gathered} )$

Worked Example The matrix M representing a transformation is given by $( \begin{gathered} 1 \frac{1}{4} & 0 \\ 0 1 \frac{1}{4} \end{gathered} )$.

Describe geometrically the transformation represented by M.

Answer:

The matrix $M$ can be written as a multiple of the identity matrix, $I$

$( \begin{gathered} 1 \frac{1}{4} & 0 \\ 0 1 \frac{1}{4} \end{gathered} ) = 1 \frac{1}{4} ( \begin{gathered} 1 0 \\ 0 1 \end{gathered} )$

So the unit square is being scaled by $\frac{1}{4}$

Final answer: Enlargement by scale factor$\frac{1}{4}$with centre at the origin