Topic 4 of 8 · Form 5
Matrices and linear transformations
Addition, multiplication (not commutative), identity I, determinant and inverse.
Notes for this topic 1 file, in order
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Matrix algebra
Addition, multiplication (not commutative), identity I, determinant and inverse.
- 2×2: for A = (a b; c d), det A = ad − bc and A⁻¹ = (1/det A)(d −b; −c a).
- 3×3 determinants by cofactor expansion; inverse via the adjugate.
- (AB)⁻¹ = B⁻¹A⁻¹.
Transformations
Matrices represent rotations, reflections, enlargements, stretches and shears. det A gives the area (volume) scale factor; a negative determinant shows a reflection (orientation reversed).
- Rotation through θ anticlockwise: (cos θ −sin θ; sin θ cos θ).
- Invariant points satisfy Ax = x; invariant lines map onto themselves.
Simultaneous equations
A unique solution exists if det ≠ 0; otherwise the system is inconsistent or has infinitely many solutions (geometric interpretation with planes).
Eigenvalues and eigenvectors
Solve det(A − λI) = 0; then (A − λI)v = 0. Diagonalisation: A = PDP⁻¹, so Aⁿ = PDⁿP⁻¹.
Worked example: Find the eigenvalues and eigenvectors of A = (2 1; 1 2).
det(A − λI) = (2 − λ)² − 1 = 0 → λ = 1 or λ = 3
λ = 1: (1 1; 1 1)v = 0 → v = (1, −1); λ = 3: (−1 1; 1 −1)v = 0 → v = (1, 1)
Check yourself
- Find the inverse of (3 2; 1 1).
- A transformation has determinant −2. What does this tell you?
- Write the matrix for a rotation of 90° anticlockwise.
Show answers
- det = 1 → (1 −2; −1 3)
- Areas are doubled and orientation is reversed (includes a reflection).
- (0 −1; 1 0)
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