3.6.2 Determinant of a Transformation Matrix

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

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

Remember that the formula for finding the determinant of a matrix is given in the formula booklet!

An isosceles triangle has vertices A(3, 1), B(15, 1) and C(9, 9).

Find the area of the isosceles triangle.

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Triangle △ABC is transformed using the matrix

T = (\begin{matrix} 3 & 2 \\ - 1 & 2 \end{matrix})

. Find the area of the transformed triangle.

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Triangle △ABC is now transformed using the matrix

U = (\begin{matrix} a & - 2 \\ 3 & a^{2} \end{matrix})

where

a \in ℤ

. Given that the area of the image is twice as large as the area of the object, find the value of

a

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DP IB Maths: AI HL

3.6.2 Determinant of a Transformation Matrix