Determinant of a Transformation Matrix
What is a determinant?
- For the 2x2 matrix
- the determinant is det A = ad - bc
What does the determinant of a transformation matrix (A) represent?
- The absolute value of the determinant of a transformation matrix is the area scale factor
- Area scale factor = |det A|
- The area of the image will be product of the area of the object and |det A|
- Area of image = |det A| × Area of object
- Note the area will reduce if |det A| < 1
- If the determinant is negative then the orientation of the shape will be reversed
- For example: the shape has been reflected
How do I solve problems involving the determinant of a transformation matrix?
- Problems may involve comparing areas of objects and images
- This could be as a percentage, proportion, etc
- Missing value(s) from the transformation matrix (and elsewhere) can be deduced if the determinant of the transformation matrix is known
- Remember to use the absolute value of the determinant
- This can lead to multiple answers to equations
- Use your GDC to solve these
Examiner Tip
- Remember that the formula for finding the determinant of a matrix is given in the formula booklet!
Worked example
An isosceles triangle has vertices A(3, 1), B(15, 1) and C(9, 9).
a)
Find the area of the isosceles triangle.
b)
Triangle △ABC is transformed using the matrix . Find the area of the transformed triangle.
c)
Triangle △ABC is now transformed using the matrix where . Given that the area of the image is twice as large as the area of the object, find the value of .