Further Pure Mathematics 1 · Matrices and transformations
Understand the terms order of a matrix, square matrix, zero matrix, and equal matrices.
Carry out matrix operations including addition, subtraction, and multiplication by a scalar.
Understand when matrices are conformable for multiplication and perform matrix multiplication.
Use a calculator to perform matrix operations.
Understand the use of matrices to represent geometric transformations in the x–y plane.
Recognise that the matrix product AB represents the transformation B followed by A.
Find the matrix that represents a given transformation or sequence of transformations.
Solve problems involving invariant points and invariant lines.
Matrix Multiplication (2x2)
The product of a 2x2 matrix with another 2x2 matrix results in a 2x2 matrix. Each element of the product is found by multiplying elements of a row from the first matrix by elements of a column from the second matrix and summing the products.
Rotation matrix (anticlockwise about origin)
This matrix transforms a point (x, y) to its image (x', y') after an anticlockwise rotation of angle θ about the origin.
Stretch parallel to x-axis
This matrix represents a stretch of scale factor m parallel to the x-axis. The y-coordinates remain unchanged.
Stretch parallel to y-axis
This matrix represents a stretch of scale factor n parallel to the y-axis. The x-coordinates remain unchanged.
Shear (x-axis fixed)
This matrix represents a shear with the x-axis fixed. Points move parallel to the x-axis.
Shear (y-axis fixed)
This matrix represents a shear with the y-axis fixed. Points move parallel to the y-axis.
Reflection in x-axis
This matrix reflects a point across the x-axis.
Reflection in y-axis
This matrix reflects a point across the y-axis.
Reflection in line y=x
This matrix reflects a point across the line y=x.
Reflection in line y=-x
This matrix reflects a point across the line y=-x.
Enlargement (centre origin, scale factor k)
This matrix represents an enlargement with the origin as the centre and a scale factor of k.
Students often think matrix multiplication is commutative (AB = BA), but actually it is generally not commutative.
Students often think any two matrices can be multiplied, but actually they must be conformable for multiplication (inner dimensions must match).
Students often confuse invariant lines with lines of invariant points; an invariant line only requires points to map to other points on the same line, not necessarily to themselves.
Students often think scalar multiplication only applies to certain elements, but actually it applies to every single element in the matrix.
Students often reverse the order of transformations when forming a composite matrix (e.g., for B followed by A, they might write AB instead of BA).