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Inverse transformations and area scale factor

Edexcel IAL Further Mathematics · WFM01 Further Pure Mathematics 1

The inverse matrix undoes a transformation, which lets you work back from an image to the original point. The determinant tells you how areas change.

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Inverse transformations

If M maps P to P', then M-1 maps P' back to P. The inverse of a combined transformation QP is P-1Q-1: undo the last transformation first.

A singular matrix (determinant 0) squashes the plane onto a line or a point, so it has no inverse: different points have the same image.

Area scale factor

If a shape has area A, its image under M has area |det M| × A. The unit square maps to a parallelogram of area |det M|.

The sign of the determinant

A negative determinant means the orientation is reversed (the vertices go the other way round), as in a reflection. Rotations have determinant 1, reflections -1, enlargement scale factor k has k2, and a stretch with scale factor k has k.

Worked example

T = (3 1; 1 2) maps P to (7, 9). Find P.

  1. det T = 6 - 1 = 5, so T-1 = (1/5)(2 -1; -1 3).
  2. P = (1/5)(2×7 - 9, -7 + 3×9) = (1/5)(5, 20).

Answer: P = (1, 4)

Key idea: Work back with M-1. Image area = |det M| × object area; negative det means orientation reversed.

Check yourself

  1. T = (3 1; 1 2) maps the point P to (7, 9). Find the y-coordinate of P.
    Show answer

    4. P = T-1(7, 9) = (1/5)(5, 20) = (1, 4).

  2. A triangle of area 6 is transformed by (2 5; 1 4). Find the area of its image.
    Show answer

    18. det = 8 - 5 = 3, so the image area is 3 × 6 = 18.

  3. A shape of area 5 is transformed by (1 4; 2 3). Find the area of the image.
    Show answer

    25. det = 3 - 8 = -5. Area scale factor |-5| = 5, so the area is 25.

  4. A transformation matrix has determinant -3. What does this tell you?
    • Areas are multiplied by -3
    • Areas are multiplied by 3 and orientation is reversed
    • Areas are divided by 3
    • The matrix is singular
    Show answer

    Areas are multiplied by 3 and orientation is reversed. |det| = 3 is the area scale factor; the negative sign shows a reflection-like reversal.

  5. Find the determinant of the matrix for any rotation about O.
    Show answer

    1. det(cos θ -sin θ; sin θ cos θ) = cos2 θ + sin2 θ = 1.

  6. P is applied then Q. Which matrix undoes the combined transformation?
    • Q-1P-1
    • P-1Q-1
    • QP
    • PQ
    Show answer

    P-1Q-1. The combined matrix is QP and (QP)-1 = P-1Q-1: undo Q first, then P.

  7. (k 2; 3 k) maps the unit square to a parallelogram of area 6, and k > 0. Find k, as a simplified surd.
    Show answer

    2√3. det = k2 - 6. Area 6 means |k2 - 6| = 6, so k2 = 12 (k2 = 0 would make k = 0). k = sqrt(12) = 2sqrt(3).

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Last updated 8 November 2026. Spotted a mistake? Tell us.