FM2: Matrices and Linear Transformations

Year 1 / ASYear 2 / A-Level All Boards (AQA, Edexcel, OCR, WJEC, CCEA) AQA

2D/3D matrix transformations, determinants, inverses, eigenvalues, eigenvectors, systems of equations.

Exam Board:

📋 Key Definitions and Core Concepts

Determinant: Scalar scale factor for area/volume. det(A) = ad - bc for 2x2.

Eigenvector: Non-zero vector v satisfying Av = λv for eigenvalue λ.

Rotation by θ: [[cos θ, -sin θ], [sin θ, cos θ]]. Reflection in y = x tan θ: [[cos 2θ, sin 2θ], [sin 2θ, -cos 2θ]].
System of 3 planes Ax = b: unique solution if det A ≠ 0; inconsistent (prism) or line of solutions (sheaf) if det A = 0.
Diagonalisation: A = PDP⁻¹ where D is diagonal eigenvalue matrix and P is eigenvector matrix.

Exam-Style Question

Question:

Find eigenvalues of A = [[4, 1], [2, 3]].

Model Solution & Mark Scheme:

det(A - λI) = (4 - λ)(3 - λ) - 2 = λ² - 7λ + 10 = (λ - 2)(λ - 5) = 0.
Eigenvalues: λ₁ = 2, λ₂ = 5.

Q1: Find inverse of [[1, 2], [3, 4]].

Show Model Answer

Answer: det = -2. Inverse = -½ [[4, -2], [-3, 1]] = [[-2, 1], [1.5, -0.5]].

Q2: Explain geometric meaning of det(M) = -1.

Show Model Answer

Answer: Area scale factor is 1, but orientation is reversed (reflection).

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