IB DP Maths: AI HL复习笔记3.6.2 Determinant of a Transformation Matrix
Determinant of a Transformation Matrix
What is a determinant?
-
- 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.
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