Matrices and Geometric Transformations
Matrices and geometric transformations uses 2 by 2 matrices to represent reflections, rotations and enlargements centred on the origin, combined by multiplying them in the correct order for one transformation followed by another. It also covers the determinant, which gives the area scale factor and identifies singular matrices, assessed for up to 8 marks in the AQA Level 2 paper.
Before you start
Make sure you're comfortable with these topics first:
Method
- Learn the standard transformation matrices: reflection in the x-axis [[1,0],[0,-1]], reflection in the y-axis [[-1,0],[0,1]], reflection in y = x [[0,1],[1,0]], rotation 90 degrees anticlockwise about the origin [[0,-1],[1,0]], and enlargement centre the origin scale factor k, [[k,0],[0,k]].
- To transform a point, write it as a column vector and multiply the transformation matrix by that vector, with the matrix on the left and the point on the right.
- To combine two transformations into a single matrix, multiply the individual matrices together, putting the matrix of the SECOND transformation on the left of the matrix of the FIRST transformation.
- Find the determinant of a 2 x 2 matrix [[a,b],[c,d]] using ad - bc; this equals the area scale factor of the transformation, ignoring sign.
- A matrix is singular if its determinant is 0; singular transformations collapse the plane onto a line and cannot be inverted.
- To find the inverse of a non-singular 2 x 2 matrix, swap the leading diagonal entries, negate the other two, and divide every entry by the determinant.
Worked example
Triangle T has area 8 cm^2 and is transformed by the matrix A = [[3, 1], [2, 4]]. Find the area of the image of T after the transformation.
- Find det(A) = (3)(4) - (1)(2) = 12 - 2 = 10.
- The area of the image = original area x |det(A)|.
- Substitute: area of image = 8 x 10 = 80.
Practice questions
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Q1Write down the matrix that represents a reflection in the x-axis.Show answer
Answer: [[1, 0], [0, -1]]
Q2Write down the matrix that represents a rotation of 180 degrees about the origin.Show answer
Answer: [[-1, 0], [0, -1]]
Q3Find the determinant of the matrix [[7, 3], [2, 5]].Show answer
Answer: 29 (7 x 5 - 3 x 2)
Q4Point P has coordinates (4, -3). Find the image of P under the transformation represented by [[0, -1], [1, 0]].Show answer
Answer: (3, 4)
Q5A matrix N = [[4, 6], [2, 3]] has determinant zero. Explain what this tells you about the transformation N.Show answer
Answer: N is singular, so it collapses the plane onto a line and has no inverse
Q6Find the matrix representing a rotation of 180 degrees about the origin followed by a reflection in the line y = x, using R = [[-1,0],[0,-1]] for the rotation and Q = [[0,1],[1,0]] for the reflection.Show answer
Answer: [[0, -1], [-1, 0]], which represents a reflection in the line y = -x
Exam-style questions
Written in the style of a GCSE Further Maths exam paper, with a full mark scheme.
Triangle T has vertices A(1, 2), B(3, 2) and C(3, 5). Find the coordinates of the image of triangle T after a reflection in the line y = x, followed by an enlargement centre the origin scale factor 2.
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E = [[4, -1], [6, 2]]. (a) Find det(E). (b) Hence state, with a reason, whether E is singular.
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Under a transformation represented by matrix T = [[p, q], [r, s]], the point (1, 0) maps to (3, -2) and the point (0, 1) maps to (4, 1). Find the matrix T.
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See real GCSE Further Maths past-paper questions, with official mark schemes →
Free printable worksheet
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