Topic 2 of 6 · Form 5
Further Pure 1: matrices and polar coordinates
2×2 and 3×3 matrices; determinants and inverses; transformations in 2D (rotations, reflections, enlargements, stretches, shears) and their compositions; invariant points and invariant lines; area scale factor |det|.
Notes for this topic 1 file, in order
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Matrices
2×2 and 3×3 matrices; determinants and inverses; transformations in 2D (rotations, reflections, enlargements, stretches, shears) and their compositions; invariant points and invariant lines; area scale factor |det|.
Worked example: Find the invariant line through the origin of the transformation M = (3 1; 2 2).
Lines y = mx map to themselves: 2x + 2mx = m(3x + mx) → m² + m − 2 = 0 → m = 1 or m = −2 → y = x and y = −2x
Systems of equations
Interpretation with planes (unique point, sheaf, prism, inconsistent).
Polar coordinates
Sketching r = f(θ); conversion between polar and Cartesian; area A = ½∫r² dθ; maximum distance from the pole.
Worked example: Find the area enclosed by r = 3 + cos θ for 0 ≤ θ ≤ 2π.
½∫(9 + 6cos θ + cos²θ) dθ = ½(18π + π) = 19π/2
Check yourself
- Find det(2 1; 5 3) and the inverse.
- What does a determinant of −3 tell you about a transformation?
- Convert r = 2 cos θ to Cartesian form.
Show answers
- 1; (3 −1; −5 2)
- Areas are multiplied by 3 and orientation is reversed.
- (x − 1)² + y² = 1
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