IBMaths: AI HLDPRevision Notes3. Geometry & Trigonometry3.6 Matrix Transformations3.6.2 Determinant of a Transformation Matrix

Determinant of a Transformation Matrix (DP IB Maths: AI HL)

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Determinant of a Transformation Matrix

What is a determinant?

  • For the 2x2 matrix bold italic A equals open parentheses table row a b row c d end table close parentheses
    • 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

Exam 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.

3-6-2-ib-hl-ai-only-we1a-soltn

b)
Triangle △ABC is transformed using the matrix bold italic T equals open parentheses table row 3 2 row cell negative 1 end cell 2 end table close parentheses. Find the area of the transformed triangle.

3-6-2-ib-hl-ai-only-we1b-soltn

c)
Triangle △ABC is now transformed using the matrix bold italic U equals open parentheses table row a cell negative 2 end cell row 3 cell a squared end cell end table close parentheses where a element of straight integer numbers. Given that the area of the image is twice as large as the area of the object, find the value of a.

3-6-2-ib-hl-ai-only-we1c-soltn

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Author: Paul

Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.