Reflection Matrices (Edexcel IGCSE Maths)

Spec point — 5G

Reflection matrices

How do I find reflection matrices?

• Imagine the unit square OABC

  • It has a side-length 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 a reflection about an axis (or y = ± x), A moves to A' and C moves to C'

  • The matrix, M representing this reflection 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 reflected square using just A' and C' (as O won't move)

• For example:

  • To find the matrix representing a reflection about the x-axis

    • A stays where it is, so $A ' = ( \begin{gathered} 1 \\ 0 \end{gathered} )$
    • C goes to $C ' = ( \begin{gathered} 0 \\ - 1 \end{gathered} )$ (on the negative y-axis)
    • $M = ( \begin{gathered} A ' \vert & C ' \end{gathered} ) = ( \begin{gathered} 1 0 \\ 0 - 1 \end{gathered} )$

  • To find the matrix representing a reflection in the line y = x

    • A goes to $A ' = ( \begin{gathered} 0 \\ 1 \end{gathered} )$ (on the positive y-axis)

    • C goes to $C ' = ( \begin{gathered} 1 \\ 0 \end{gathered} )$ (on the positive x-axis)

    • $M = ( \begin{gathered} A ' \vert & C ' \end{gathered} ) = ( \begin{gathered} 0 1 \\ 1 0 \end{gathered} )$

      • This is not the same as the identity matrix as the 1s are on the wrong diagonal

Worked Example (a) The matrix M represents a reflection in the y-axis. Work out M.

Answer:

Consider how the points A and C on the unit square are transformed by a reflection in the y-axis

The point A $( \begin{gathered} 1 \\ 0 \end{gathered} )$ moves to A' $( \begin{gathered} - 1 \\ 0 \end{gathered} )$* *

The point C $( \begin{gathered} 0 \\ 1 \end{gathered} )$ remains in the same place

The transformation matrix is given by $M = ( \begin{gathered} A ' \vert & C ' \end{gathered} )$* *

Final answer: $\mathrm{M} = ( \begin{gathered} - 1 & 0 \\ 0 1 \end{gathered} )$

(b) The matrix N represents a reflection in the line $y=-x$. Work out N.

Answer:

Consider how the points A and C on the unit square are transformed by a reflection in the line $y=-x$

The point A $( \begin{gathered} 1 \\ 0 \end{gathered} )$ moves to A' $( \begin{gathered} 0 \\ - 1 \end{gathered} )$* *

The point C $( \begin{gathered} 0 \\ 1 \end{gathered} )$ moves to C' $( \begin{gathered} - 1 \\ 0 \end{gathered} )$

The transformation matrix is given by $N = ( \begin{gathered} A ' \vert & C ' \end{gathered} )$* *

$\mathrm{N} = ( \begin{gathered} 0 - 1 \\ - 1 & 0 \end{gathered} )$