Matrices and Linear Transformations

🎯 Learning Objectives

• Multiply conformable 2x2 matrices and interpret a matrix product as a composition of transformations, remembering AB ≠ BA (AO1) - Edexcel 9FM0/01 Core Pure 1; OCR MEI H645 Further Pure
• Compute 2x2 determinants and use det(AB) = det A × det B, including the determinant as an area scale factor (AO1/AO2)
• Find the inverse of a non-singular 2x2 matrix and use (AB)⁻¹ = B⁻¹A⁻¹ (AO1)
• Solve systems of linear equations using matrix inverses and interpret singular cases as no solution or infinitely many solutions (AO2/AO3)
• Write down matrices for rotations, reflections and enlargements and combine them in the correct order (AO1/AO2)
• Find invariant points and invariant lines of a linear transformation, including lines of invariant points for reflections (AO2/AO3)

📋 Key Definitions and Core Concepts

📌 Key Points

Matrix Arithmetic and Determinants

Products. For 2x2 matrices A = [[a, b], [c, d]] and B = [[e, f], [g, h]], the product AB has (i, j) entry given by multiplying row i of A into column j of B: AB = [[ae + bg, af + bh], [ce + dg, cf + dh]]. Multiplication is associative (A(BC) = (AB)C) and distributes over addition, but is not commutative: in general AB ≠ BA. The identity I = [[1, 0], [0, 1]] satisfies AI = IA = A. When a question gives a matrix equation such as A² = A (idempotent) or asks you to find matrix X satisfying AX = B, rearrange by multiplying by inverses on the correct side - left-multiplication and right-multiplication are different operations.

Worked example. A = [[2, 1], [5, 3]], B = [[0, 2], [3, 1]]: AB = [[2×0 + 1×3, 2×2 + 1×1], [5×0 + 3×3, 5×2 + 3×1]] = [[3, 5], [9, 13]]. Note BA = [[10, 6], [11, 6]] - different, as expected.

Determinants. det M = ad − bc, also written det(M) or |M|. Two properties carry most of the mark weight: det(AB) = det A × det B, and det(Aᵀ) = det A. In the example above, det A = 2×3 − 1×5 = 1 and det B = 0×1 − 2×3 = −6, so det(AB) = −6 - and indeed 3×13 − 5×9 = 39 − 45 = −6 ✓. Exam questions frequently ask you to verify this product rule.

Geometric meaning. For a transformation with matrix M, the image of a shape has area multiplied by |det M|, and det M < 0 signals that orientation is reversed (a flip). If det M = 0 the matrix is singular: the transformation squashes the whole plane onto a line (or a point), so distinct points share images and no inverse can exist. Singular matrices are the gateway to the system-of-equations cases in the next section.

Inverses and Systems of Equations

The inverse formula. If det M = ad − bc ≠ 0, then M⁻¹ = (1/(ad − bc))[[d, −b], [−c, a]]: swap the leading diagonal, negate the other diagonal, divide by the determinant. Always check MM⁻¹ = I if asked to verify. The reversal rule (AB)⁻¹ = B⁻¹A⁻¹ holds whenever both factors are invertible - order flips, exactly as for transposes. A matrix is invertible if and only if it is non-singular.

Solving linear systems. The system 2x + 3y = 13, x − 2y = −4 is Mx = b with M = [[2, 3], [1, −2]] and b = (13, −4)ᵀ. det M = −4 − 3 = −7, so M⁻¹ = (1/7)[[2, 3], [1, −2]] and x = M⁻¹b = (1/7)[[2, 3], [1, −2]](13, −4)ᵀ = (1/7)(26 − 12, 13 + 8)ᵀ = (2, 3)ᵀ. Substituting back: 2×2 + 3×3 = 13 ✓. State the matrix equation, the determinant, the inverse and the solution - each step typically carries a mark.

Singular cases. When det M = 0 there is no unique solution, and the two possibilities are distinguished by consistency. The system x + 2y = 3, 2x + 4y = 7 has det [[1, 2], [2, 4]] = 0; doubling the first equation predicts 2x + 4y = 6, which contradicts 7, so the lines are parallel and distinct: no solution (inconsistent). But x + 2y = 3, 2x + 4y = 6 has the second equation a multiple of the first: the two equations describe the same line, so there are infinitely many solutions (a line of solutions). For 3x3 systems - which appear in OCR MEI Further Pure - the same trichotomy holds: unique (det ≠ 0), no common point, or a line/plane of common solutions.

Linear Transformations, Invariant Points and Invariant Lines

Matrices as maps. A 2x2 matrix M acts on position vectors: the point with position vector x maps to Mx. The unit square maps to a parallelogram whose sides are the columns of M - reading off the images of (1, 0) and (0, 1) is the fastest way to identify or construct a transformation. The standard matrices must be memorised:

Check the reflection formula with y = x (α = 45°): [[cos 90°, sin 90°], [sin 90°, −cos 90°]] = [[0, 1], [1, 0]] ✓. A negative enlargement factor k combines an enlargement with a 180° rotation, and has det k² > 0. Determinants classify the map: rotation det = 1 (area preserved), reflection det = −1 (area preserved, orientation reversed), enlargement det = k².

Composites. The matrix of transformation T followed by S is ST - the matrix acting first sits on the right. With Rθ denoting rotation by θ, note that RθRφ = R(θ+φ) but a reflection followed by a rotation is generally not a rotation: order matters both for the type of map and for its parameters.

Invariant points and lines. A point is invariant when Mx = x, i.e. (M − I)x = 0. An invariant line maps to itself as a set - points may slide along it. For a line through the origin with gradient m, require M(1, m)ᵀ = k(1, m)ᵀ. Worked example: M = [[3, −1], [−1, 3]]. Points on y = x: M(t, t)ᵀ = (2t, 2t)ᵀ - still on y = x, so y = x is an invariant line with scale factor 2 (only the origin is fixed). For general m: M(1, m)ᵀ = (3 − m, −1 + 3m)ᵀ = k(1, m)ᵀ, so k = 3 − m and −1 + 3m = m(3 − m), giving m² = 1: the invariant lines are y = x (scale 2) and y = −x (scale 4).

Structure to quote for common maps: for a reflection, the mirror line consists entirely of invariant points, and the line through O perpendicular to it is invariant with no fixed points (except O); for an enlargement with k ≠ 1, only O is an invariant point but every line through O is an invariant line.

💡 Worked Example

❓ Practice Questions & Mark Schemes

🎬 Video Resources

• YouTube: 2x2 matrices inverse determinant invariant lines linear transformations A-level Further Maths Edexcel

📄 Past Papers & Exam Resources

• Edexcel A-level Further Maths past papers (PMT)
• Edexcel past questions by topic (PMT) - Further Pure
• Pearson Edexcel A-levels - specification and assessment materials
• Pearson Edexcel A-level Further Mathematics (9FM0) qualification page

🔗 Further Reading & Resources

• PMT Further Pure 1 revision - matrices notes and worksheets
• PMT Further Pure 2 revision - notes and topic questions
• PMT Maths A-level revision hub

🧠 Flashcards (Spaced Repetition)

📝 Exam Questions by Topic

🎯 Target Tests (Auto-Graded)

🎓 Smart Lesson (Guided)