Transformations using Matrices (Edexcel IAL Further Maths)

Transforming Points using Matrices

Spec point — 6.2

How do I transform a point using a matrix?

• A point $( x , y )$ in a 2D plane can be transformed (mapped) on to another point $( x ' , y ' )$ by a 2 × 2 matrix, $M$

  • This is called a linear transformation

    • $( x , y )$ is the object and $( x ' , y ' )$ is the image

• The coordinates of the image point can be found using matrix multiplication

• To transform $( x , y )$ by the matrix $( \begin{gathered} a b \\ c d \end{gathered} )$

  • Write $( x , y )$ as a column vector, $( \begin{gathered} x \\ y \end{gathered} )$
  • Use matrix multiplication to work out $( \begin{gathered} a b \\ c d \end{gathered} ) ( \begin{gathered} x \\ y \end{gathered} )$, which gives $( \begin{gathered} x ' \\ y ' \end{gathered} )$
  • Write down the image point coordinates, $( x ' , y ' )$

• If given image coordinates, $( x ' , y ' )$, and asked to find original coordinates

  • use inverse matrices
  • $M ( \begin{gathered} x \\ y \end{gathered} ) = ( \begin{gathered} x ' \\ y ' \end{gathered} ) \Leftrightarrow ( \begin{gathered} x \\ y \end{gathered} ) = M^{- 1} ( \begin{gathered} x ' \\ y ' \end{gathered} )$

How do I transform a set of vertices?

• Vertices of a shape can be transformed one-by-one using the method above

  • This gives the vertices of the new shape

    • Straight lines between the vertices will still be straight lines in the new shape

  • Sometimes the order (clockwise or anticlockwise) of the vertices is reversed

    • This is also called changing the **sense **of the vertices

• An** alternative method** is to stack column vectors of coordinates into a vertex matrix

  • $( 1 , 2 )$, $( 3 , 4 )$, $( 5 , 6 )$ and $( 0 , 10 )$ become $( \begin{gathered} 1 3 5 0 \\ 2 4 6 10 \end{gathered} )$
  • Then multiply the matrix $M$ by the vertex matrix above
  • The new columns represent the corresponding image coordinates

Worked Example The matrix $M$ is given by $M = ( \begin{gathered} 4 5 \\ 1 - 2 \end{gathered} )$. A triangle has coordinates $( 1 , 0 )$, $( 2 , 3 )$ and $( - 1 , 5 )$.

Work out the coordinates of the triangle after the transformation represented by the matrix $M$ has been applied.

*Multiply the matri**x $M$ by each set *of coordinates, written as a column vectors

$\begin{gathered} ( \begin{gathered} 4 5 \\ 1 - 2 \end{gathered} ) ( \begin{gathered} 1 \\ 0 \end{gathered} ) & = ( \begin{gathered} 4 \times 1 + 5 \times 0 \\ 1 \times 1 + - 2 \times 0 \end{gathered} ) = ( \begin{gathered} 4 \\ 1 \end{gathered} ) \\ ( \begin{gathered} 4 5 \\ 1 - 2 \end{gathered} ) ( \begin{gathered} 2 \\ 3 \end{gathered} ) & = ( \begin{gathered} 4 \times 2 + 5 \times 3 \\ 1 \times 2 + - 2 \times 3 \end{gathered} ) = ( \begin{gathered} 23 \\ - 4 \end{gathered} ) \\ ( \begin{gathered} 4 5 \\ 1 - 2 \end{gathered} ) ( \begin{gathered} - 1 \\ 5 \end{gathered} ) & = ( \begin{gathered} 4 \times ( - 1 ) + 5 \times 5 \\ 1 \times ( - 1 ) + - 2 \times 5 \end{gathered} ) = ( \begin{gathered} 21 \\ - 11 \end{gathered} ) \end{gathered}$

Rewrite the answers as coordinates

Final answer: $( 4 , 1 )$,$( 23 , - 4 )$and$( 21 , - 11 )$

Alternatively, you can multi ply $M$by a vertex mat rix: $( \begin{gathered} 4 5 \\ 1 - 2 \end{gathered} ) ( \begin{gathered} 1 2 - 1 \\ 0 3 5 \end{gathered} ) = ( \begin{gathered} 4 23 21 \\ 1 - 4 & - 11 \end{gathered} )$

Determinants as Area Scale Factors

Spec point — 6.4

How are 2x2 determinants and areas related?

• When transforming vertices of an object using the matrix$M$ to give vertices of the image

  • the magnitude of the determinant of $M$ is the area scale factor of enlargement from the object to the image
  • $\vert \det ( M ) \vert$is the area scale factor

• For example, if a shape is transformed by $M = ( \begin{gathered} 1 3 \\ 2 4 \end{gathered} )$ then

  • $\det ( M ) = 1 \times 4 - 2 \times 3 = - 2$

  • so$\vert \det ( M ) \vert = \vert - 2 \vert = 2$

    • This transformation doubles the area of any shape

• A negative determinant means the sense (clockwise or anticlockwise) of the vertices is reversed

Exam Hint Remember the modulus signs around the determinant as this can often lead to two solutions in algebraic questions.

Worked Example A transformation, represented by the matrix $M = ( \begin{gathered} 3 2 \\ 4 k + 1 \end{gathered} )$, where $k$ is a constant, maps a quadrilateral of area 45 square units to a quadrilateral of area 450 square units.

Find the possible values of $k$.

Find the scale factor of enlargement of the area from 45 to 450

Area scale factor is 10

The magnitude of the determinant is the area scale factor Start by finding and simplifying the determinant of $M$

$\begin{gathered} \det ( M ) & = 3 ( k + 1 ) - 2 \times 4 \\ = 3 k + 3 - 8 \\ = 3 k - 5 \end{gathered}$

Set the magnitude of the determinant equal to 10

$\vert 3 k - 5 \vert = 10$

This means $3k-5$ is either equal to 10 or -10 Solve each equation

$\begin{gathered} 3 k - 5 & = 10 \\ 3 k & = 15 \\ k = 5 \end{gathered}$

or

$\begin{gathered} 3 k - 5 & = - 10 \\ 3 k & = - 5 \\ k = - \frac{5}{3} \end{gathered}$

Write out the two answers

$k=5\mathrm{or}k=-\frac{5}{3}$

If the question had said "where$k$is an integer" then only$k=5$would be accepted