Determinant of a Transformation Matrix (Edexcel IGCSE Maths)

Spec point — 5G

Determinant of a transformation matrix

What does the determinant of a transformation matrix represent?

• When a 2×2 matrix is used to represent a transformation

  • the determinant of the matrix is the area scale factor of the transformation

  • I.e. the area of a transformed shape is equal to

    • the area of the original shape
    • times the determinant of the transformation matrix

• The determinant of a 2×2 matrix $( \begin{gathered} a b \\ c d \end{gathered} )$ is $ad-bc$

Worked Example A triangle has vertices with coordinates (1, -3), (7, -3) and (3, 1).

(a) Find the coordinates of the vertices of the image triangle when that triangle is transformed by the matrix $M = ( \begin{gathered} - \sqrt{3} & - 1 \\ 1 - \sqrt{3} \end{gathered} )$.

Answer:

You can represent the triangle as a 2×3 matrix of vertices

• This will be the coordinates of the vertices as column vectors, but written in a single matrix

$( \begin{gathered} 1 7 3 \\ - 3 & - 3 & 1 \end{gathered} )$

To transform the triangle, multiply its matrix by the transformation matrix $M$

$\begin{gathered} ( \begin{gathered} - \sqrt{3} & - 1 \\ 1 - \sqrt{3} \end{gathered} ) ( \begin{gathered} 1 7 3 \\ - 3 & - 3 & 1 \end{gathered} ) \\ = ( \begin{gathered} ( - \sqrt{3} ) \times 1 + ( - 1 ) \times ( - 3 ) & ( - \sqrt{3} ) \times 7 + ( - 1 ) \times ( - 3 ) & ( - \sqrt{3} ) \times 3 + ( - 1 ) \times 1 \\ 1 \times 1 + ( - \sqrt{3} ) \times ( - 3 ) & 1 \times 7 + ( - \sqrt{3} ) \times ( - 3 ) & 1 \times 3 + ( - \sqrt{3} ) \times 1 \end{gathered} ) \\ = ( \begin{gathered} 3 - \sqrt{3} & 3 - 7 \sqrt{3} & - 1 - 3 \sqrt{3} \\ 1 + 3 \sqrt{3} & 7 + 3 \sqrt{3} & 3 - \sqrt{3} \end{gathered} ) \end{gathered}$

Write the new vertices as coordinates

Final answer: $( 3 - \sqrt{3} , 1 + 3 \sqrt{3} )$,$( 3 - 7 \sqrt{3} , 7 + 3 \sqrt{3} )$and$( - 1 - 3 \sqrt{3} , 3 - \sqrt{3} )$

(b) Find the area of the image triangle.

Answer:

The original and transformed triangles look like this:

Trying to calculate the area of the transformed triangle directly would be quite challenging, to say the least!

However the area of the original triangle is easy to calculate

• It has a base of 6, and a height of 4
• Use $\mathrm{Area}=\frac{1}{2}\times\mathrm{base}\times\mathrm{height}$

$\mathrm{area} \mathrm{of} \mathrm{original} \mathrm{triangle}=\frac{1}{2}\times6\times4=12$

Find the determinant of the transformation matrix $M = ( \begin{gathered} - \sqrt{3} & - 1 \\ 1 - \sqrt{3} \end{gathered} )$

• *The determinant of a 2×2 matrix *$( \begin{gathered} a b \\ c d \end{gathered} )$ is $ad-bc$

| determinant of M | = ( - sqrt( 3 ) x ( - sqrt( 3 ) - ( - 1 ) x 1 | | _ = 3 - ( - 1 ) | | _ = 4 |

That is the area scale factor of the transformation

• So multiply the original area by 4 to find the transformed area

$12\times4=48$

Final answer: Area of the image triangle is 48