Standard Matrix Transformations (Edexcel IAL Further Maths)

Reflection Matrices

Spec point — 6.2

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$y=\pm x$, 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 their new positions

    • The points O * and B are not needed, as we can draw the reflected square using just A' and C' (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 $M$.

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

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

(b) Describe fully the transformation represented by the matrix $N = ( \begin{gathered} 0 - 1 \\ - 1 & 0 \end{gathered} )$.

*Consider how the points A and C on the unit square are transformed *

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' *$( \begin{gathered} - 1 \\ 0 \end{gathered} )$

It helps to draw a picture of the unit square being transformed with vertices clearly labelled

*This transformation could be a rotation of 180 ° about O or a reflection in $y=-x$ * The vertices A' and C' are in the correct places for a reflection, but not a rotation

Final answer: The matrix N represents a reflection in the line$y=-x$

Enlargement & Stretch Matrices

Spec point — 6.2

Which matrix represents an enlargement?

• The matrix $M = ( \begin{gathered} k 0 \\ 0 k \end{gathered} )$ represents an enlargement of scale factor$k$ about the origin, O

  • This is the same as $M=kI$

    • $I$ is the identity matrix

Which matrix represents a stretch?

• The matrix $M = ( \begin{gathered} a 0 \\ 0 1 \end{gathered} )$ represents a stretch parallel to the$x$-axis of scale factor$a$

  • The point $( x , y )$ becomes $( a x , y )$

• The matrix $M = ( \begin{gathered} 1 0 \\ 0 b \end{gathered} )$ represents a stretch parallel to the$y$-axis of scale factor$b$

  • The point $( x , y )$ becomes $( x , b y )$

• The matrix $M = ( \begin{gathered} a 0 \\ 0 b \end{gathered} )$ represents a combined stretch of scale factor$a$ parallel to the $x$-axis and scale factor$b$parallel to the $y$-axis

  • If $a=b$, the combined stretch is an enlargement

Exam Hint Use phrases like "parallel to the $x$-axis" or "parallel to the $y$-axis" to describe stretches (not "left" or "up"!)

Worked Example A transformation is represented by the matrix $M = ( \begin{gathered} 3 + p & 0 \\ 0 3 - p \end{gathered} )$.

Describe fully the transformation in each of the following cases:

(a) $p=0$

Substitute in $p=0$

$M = ( \begin{gathered} 3 + 0 & 0 \\ 0 3 - 0 \end{gathered} ) = ( \begin{gathered} 3 0 \\ 0 3 \end{gathered} )$

This has the form $M = ( \begin{gathered} k 0 \\ 0 k \end{gathered} )$ where $k=3$

Final answer: $M$represents an enlargement of scale factor 3 about the origin

You must give its scale factor and centre of enlargement

(b) $p=2$

Substitute in $p=2$

$M = ( \begin{gathered} 3 + 2 & 0 \\ 0 3 - 2 \end{gathered} ) = ( \begin{gathered} 5 0 \\ 0 1 \end{gathered} )$

This has the form $M = ( \begin{gathered} a 0 \\ 0 1 \end{gathered} )$ where $a=5$

Final answer: $M$represents a stretch of scale factor 5 parallel to the$x$-axis

You must give its scale factor and direction

Rotation Matrices

Spec point — 6.2

How do I find matrices for rotations by multiples of 90°?

• 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 a rotation about the origin, A moves to A' and C moves to C'

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

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

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

• For example:

  • To find the matrix representing a rotation of 90° anticlockwise about the origin

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

  • To find the matrix representing a rotation of 180° about the origin

    • A goes to $A ' = ( \begin{gathered} - 1 \\ 0 \end{gathered} )$ (on the negative $x$-axis)
    • C goes to $C ' = ( \begin{gathered} 0 \\ - 1 \end{gathered} )$ (on the negative $y$-axis)
    • $M = ( \begin{gathered} A ' \vert & C ' \end{gathered} ) = ( \begin{gathered} - 1 & 0 \\ 0 - 1 \end{gathered} )$
    • This is the same as $M=-I$ where* *$I$ is the identity matrix

Exam Hint Students often confuse rotations of 180° with reflections in the lines $y=\pm x$ .

How do I find matrices for rotations by any angle?

• A rotation anticlockwise by any angle, $θ$, about the origin is represented by the matrix:

  • $M = ( \begin{gathered} \cos \\theta & - \sin \\theta \\ \sin \\theta & \cos \\theta \end{gathered} )$

    • $θ$ can be in degrees or radians

• A negative value of $θ$ represents a clockwise rotation

  • Remember that $\sin ( - \\theta ) = - \sin \\theta$ but that $\cos ( - \\theta ) = \cos \\theta$

• You can substitute in multiples of 90° to get the matrices above

• You may be required to recognise common angles from their ratios

  • For example, (sqrt( 3)/(2) = cos ( pi / 6 )

Exam Hint You are given the rotation matrix in the Formulae Booklet.

Worked Example (a) Describe fully the transformation represented by the matrix $P = ( \begin{gathered} 0 1 \\ - 1 & 0 \end{gathered} )$.

*Consider how the points A and C on the unit square are transformed *

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' *$( \begin{gathered} 1 \\ 0 \end{gathered} )$

It helps to draw a picture of the unit square being transformed with vertices clearly labelled

This transformation could be a rotation of 90 ° clockwise about O or a reflection in the $x$-axis The vertices A' and C' are not in the correct places for a reflection, but are for a rotation

The matrix P represents a rotation of 90°clockwise about the origin

You must give its angle, direction and centre of rotation 270°anticlockwise would also be accepted

(b) Find $Q$, the matrix that represents a clockwise rotation of 120° about the origin, giving your answer in an exact form.

The matrix for an anticlockwise rotation by $θ$ is$( \begin{gathered} \cos \\theta & - \sin \\theta \\ \sin \\theta & \cos \\theta \end{gathered} )$ $θ$ is negative, as the rotation is clockwise

$θ=-120^{\circ}$

Substitute this value of $θ$ into the matrix Use that $\cos ( - \\theta ) = \cos \\theta$ and that $\sin ( - \\theta ) = - \sin \\theta$

$( \begin{gathered} \cos ( - 120^{\circ} ) & - \sin ( - 120^{\circ} ) \\ \sin ( - 120^{\circ} ) & \cos ( - 120^{\circ} ) \end{gathered} ) = ( \begin{gathered} \cos ( 120^{\circ} ) & \sin ( 120^{\circ} ) \\ - \sin ( 120^{\circ} ) & \cos ( 120^{\circ} ) \end{gathered} )$

Use a calculator to find these values (or common angles and symmetry)

$Q = ( \begin{gathered} - 1 \frac{1}{2} & \frac{ \sqrt{3} }{2} \\ - \frac{ \sqrt{3} }{2} & - 1 \frac{1}{2} \end{gathered} )$