Vectors and Matrices
Vectors and matrices are grouped because both are linear tools. The paper tests vector geometry in two and three dimensions, the scalar product, equations of lines, and 2×2 (occasionally 3×3) matrix algebra including determinants, inverses and geometric transformations.
Vectors
- Magnitude |a| = √(a₁² + a₂² + a₃²); a unit vector is a/|a|.
- Scalar product a·b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃; a·b = 0 exactly when the vectors are perpendicular.
- The line through position vector a with direction b is r = a + tb.
- Two lines meet when a + tb = c + sd has a consistent solution; parallel directions with inconsistent positions give distinct parallel lines.
Matrices
- AB is defined when the column count of A equals the row count of B; in general AB ≠ BA.
- det[[a, b], [c, d]] = ad − bc; A⁻¹ = (1/det)[[d, −b], [−c, a]], existing only when det ≠ 0.
- (AB)⁻¹ = B⁻¹A⁻¹ - reverse the order.
- Transformations: anticlockwise rotation θ is [[cos θ, −sin θ], [sin θ, cos θ]]; reflection in y = x is [[0, 1], [1, 0]]; enlargement factor k is kI.
- Simultaneous equations AX = B are solved by X = A⁻¹B.
Worked example
Solve 3x + 2y = 7 and x + 4y = 9. Here A = [[3, 2], [1, 4]] with det = 12 − 2 = 10, so A⁻¹ = (1/10)[[4, −2], [−1, 3]]. Then X = A⁻¹B = (1/10)[[4, −2], [−1, 3]][[7], [9]] = (1/10)[[10], [20]] = [[1], [2]]. So x = 1, y = 2, and substitution confirms both equations.
Traps
- Multiplying matrices in the wrong order, or writing A⁻¹B as BA⁻¹.
- Inverting A and B separately but forgetting to reverse their order in a product.
- Ignoring the singular case, where the equations are inconsistent or admit infinitely many solutions.