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A-Level Further Mathematics revision: Matrices. Learning objectives, key points, worked examples and practice questions across AQA, Edexcel, OCR, WJEC and CCEA.
📌 Key Points
Key Fact: Multiplication: (AB)_{ij} = Σₖ A_{ik}B_{kj} -- non-commutative, associative, distributive
Key Fact: det 2x2: |a b; c d| = ad - bc; det 3x3: cofactor expansion or Sarrus
Key Fact: Inverse 2x2: A⁻¹ = (1/det)[d -b; -c a]; 3x3: adj(A)/det(A) via minors/cofactors
Key Fact: System AX = B: if det(A) != 0, X = A⁻¹B (unique solution); if det(A) = 0, either no solution or infinite
Key Fact: Transformations: reflection in x-axis [1 0; 0 -1], rotation by θ [cosθ -sinθ; sinθ cosθ]
Key Fact: Eigenvalues λ: det(A - λI) = 0; Eigenvectors: (A - λI)v = 0
Key Fact: Diagonalisation: P⁻¹AP = D (D diagonal) if A has n linearly independent eigenvectors
🎯 Learning Objectives
- Perform matrix addition, subtraction and multiplication
- Find determinants of 2x2 and 3x3 matrices
- Find inverses of 2x2 and 3x3 matrices using adjugate or row reduction
- Solve systems of linear equations using inverse matrices
- Understand singular matrices and consistency of systems
- Apply matrices to geometric transformations (reflection, rotation, enlargement)
- Find eigenvalues and eigenvectors of 2x2 matrices
- Diagonalise symmetric matrices
💡 Worked Example
Exam-Style Question
Question: Find eigenvalues and eigenvectors of A = [3 2; 1 4]. Diagonalise A
Model Answer:
det(A-λI) = (3-λ)(4-λ)-2 = λ^2-7λ+10 = (λ-5)(λ-2)=0 -> λ=5,2. λ=5: [−2 2; 1 −1]v=0 -> v₁=[1;1]. λ=2: [1 2; 1 2]v=0 -> v₂=[2;−1]. P=[1 2; 1 −1], D=[5 0; 0 2]. Check P⁻¹AP=D
❓ Practice Questions
Model answers are being added progressively - questions marked ✗ don't have one yet. Cross-check with your teacher or the official mark scheme.
Questions:
- Find det and inverse of [2 1 3; 0 1 4; 5 6 0]✗ answer coming soon
- Solve using matrix inverse: 2x+y-z=1, x+2y+z=2, 3x-y+2z=3✗ answer coming soon
- Find matrix for rotation by 90 deg about origin followed by reflection in y=x✗ answer coming soon
- Find eigenvalues of [4 1; 2 3] and corresponding eigenvectors✗ answer coming soon
- Show [1 2; 2 1] is diagonalisable✗ answer coming soon
🎬 Video Resources
- Matrices - Overview -- YouTube Search
- Matrices - Exam Questions -- YouTube Search
📄 Past Papers & Exam Resources
🔗 Further Reading & Resources
- Physics & Maths Tutor - Matrices -- Free notes, worksheets, and past paper questions by topic
- Dr Frost Maths - Matrices -- Free resources, slides, and interactive questions
- BBC Bitesize - Matrices -- BBC Bitesize revision resources
📚 Lesson Plan (50 minutes)
- Starter (5 min): Recall prior knowledge of matrices with quick questions.
- Teaching (15 min): Work through each of the learning objectives, explaining principles step by step.
- Key points review (5 min): Revisit the key points together, confirming understanding.
- Worked example (10 min): Model the example question: Find eigenvalues and eigenvectors of A = [3 2; 1 4]. Diagonalise A. Solution: det(A-λI) = (3-λ)(4-λ)-2 = λ^2-7λ+10 = (λ-5)(λ-2)=0 -> λ=5,2. λ=5: [−2 2; 1 −1]v=0 -> v₁=[1;1]. λ=2: [1 2; 1 2]v=0 -> v₂=[2;−1]. P=[1 2; 1 −1], D=[5 0; 0 2]. Check P⁻¹AP=D
- Practice (10 min): Students attempt the practice questions independently; circulate and support.
- Plenary (5 min): Review answers and address misconceptions.
🏠 Homework
- Find det and inverse of [2 1 3; 0 1 4; 5 6 0]
- Solve using matrix inverse: 2x+y-z=1, x+2y+z=2, 3x-y+2z=3
- Find matrix for rotation by 90 deg about origin followed by reflection in y=x
- Find eigenvalues of [4 1; 2 3] and corresponding eigenvectors
- Show [1 2; 2 1] is diagonalisable
🧾 Assessment
Check practice answers against the model answer; use the built-in practice questions as formative assessment.