Enlargement Matrices (Edexcel IGCSE Maths B): Revision Note

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

  • $A=\left(\begin{array}{c}1\\ 0\end{array}\right)$ and $C=\left(\begin{array}{c}0\\ 1\end{array}\right)$

• 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 $\mathbf{M}=\left(\begin{array}{cc}A\text{'}|& C\text{'}\end{array}\right)$

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

    • So $\mathbf{M}$ 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)

• $A\text{'}=\left(\begin{array}{c}k\\ 0\end{array}\right)$ and $C\text{'}=\left(\begin{array}{c}0\\ k\end{array}\right)$

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

• So all enlargement matrices have the form $\mathbf{M}=\left(\begin{array}{cc}k& 0\\ 0& k\end{array}\right)$

  • This is the same as $\mathbf{M}=k\mathbf{I}$, where $\mathbf{I}$ is the identity matrix

• For example:

  • The matrix representation of an enlargement of scale factor 3 with centre at the origin is $\left(\begin{array}{cc}3& 0\\ 0& 3\end{array}\right)$

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