Determinant of a Transformation Matrix (DP IB Applications & Interpretation (AI)): Revision Note

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