Transformations using Matrices (Edexcel International A Level Further Maths)
Revision Note
Author
Mark CurtisExpertise
Maths
Transforming Points using Matrices
How do I transform a point using a matrix?
A point in a 2D plane can be transformed (mapped) on to another point by a 2 × 2 matrix,
This is called a linear transformation
is the object and is the image
The coordinates of the image point can be found using matrix multiplication
To transform by the matrix
Write as a column vector,
Use matrix multiplication to work out , which gives
Write down the image point coordinates,
If given image coordinates, , and asked to find original coordinates
use inverse matrices
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
, , and become
Then multiply the matrix by the vertex matrix above
The new columns represent the corresponding image coordinates
Worked Example
The matrix is given by .
A triangle has coordinates , and .
Work out the coordinates of the triangle after the transformation represented by the matrix has been applied.
Multiply the matrix by each set of coordinates, written as a column vectors
Rewrite the answers as coordinates
, and
Alternatively, you can multiply by a vertex matrix:
Determinants as Area Scale Factors
How are 2x2 determinants and areas related?
When transforming vertices of an object using the matrix to give vertices of the image
the magnitude of the determinant of is the area scale factor of enlargement from the object to the image
is the area scale factor
For example, if a shape is transformed by then
so
This transformation doubles the area of any shape
A negative determinant means the sense (clockwise or anticlockwise) of the vertices is reversed
Exam Tip
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 , where is a constant, maps a quadrilateral of area 45 square units to a quadrilateral of area 450 square units.
Find the possible values of .
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
Set the magnitude of the determinant equal to 10
This means is either equal to 10 or -10
Solve each equation
or
Write out the two answers
If the question had said "where is an integer" then only would be accepted
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