Matrices

Further Maths · Further Pure Mathematics 1. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

A matrix is a block of numbers. Matrices are multiplied row by column, and the order matters: AB is usually not equal to BA. A square matrix has a determinant. If the determinant is zero the matrix is singular and has no inverse.

A 2 × 2 matrix also describes a transformation of the x-y plane that keeps the origin fixed: a rotation, reflection, enlargement, stretch or shear. Its two columns are the images of the points (1, 0) and (0, 1).

An invariant point is a point that does not move. An invariant line is a line whose points all map to points on the same line, although each point may move along it.

Rules to remember

AB exists only if (number of columns of A) = (number of rows of B). In general AB ≠ BA, but AI = IA = A
For M with rows (a, b) and (c, d): det M = ad − bc, and M⁻¹ = [1/(ad − bc)] × the matrix with rows (d, −b) and (−c, a)
Singular means det M = 0 (no inverse). Also det(AB) = det A × det B
(AB)⁻¹ = B⁻¹A⁻¹ and (ABC)⁻¹ = C⁻¹B⁻¹A⁻¹
Rotation through θ anticlockwise about the origin: rows (cos θ, −sin θ) and (sin θ, cos θ). A⁻¹ is the transformation that undoes A
AB means do B first, then A
Area of image = |det M| × area of object. A negative determinant means the shape has also been turned over

Common mistake

For 'transformation P followed by transformation Q' students write PQ. The first transformation goes on the right, next to the point, so the correct product is QP.

Worked example

M is the matrix with rows (4, −1) and (2, 1). Find the invariant lines through the origin for the transformation represented by M.

A point on y = mx is (1, m). Its image is (4 × 1 − 1 × m, 2 × 1 + 1 × m) = (4 − m, 2 + m).
The image must lie on y = mx: 2 + m = m(4 − m).
Rearrange: m² − 3m + 2 = 0, so (m − 1)(m − 2) = 0 and m = 1 or m = 2.
Check: (1, 1) → (3, 3) and (1, 2) → (2, 4), both still on their own lines.

Answer: y = x and y = 2x

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Matrices

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