Simultaneous equations by matrices — A/L Combined Mathematics

A system of two linear simultaneous equations in two variables: a₁x + b₁y = d₁ and a₂x + b₂y = d₂ can be structured in matrix form as AX = B, where A = [[a₁, b₁], [a₂, b₂]] is the coefficient matrix, X = [[x], [y]] is the variable vector, and B = [[d₁], [d₂]] is the constant vector. If det(A) = a₁b₂ - a₂b₁ ≠ 0, the matrix A is invertible, and multiplying both sides from the left by A⁻¹ yields the unique solution: X = A⁻¹B. If det(A) = 0, the system either has no solution (inconsistent, representing parallel lines) or infinitely many solutions (coincident lines).

Check before you answer (A/L Pure Maths Paper 1)

1. Matrix Equation Formulation: Correctly writing the given equations as AX = B, showing all matrix components clearly.
2. Inverse Premultiplication: Pre-multiplying by A⁻¹ explicitly (AX = B ⟹ A⁻¹(AX) = A⁻¹B ⟹ IX = A⁻¹B ⟹ X = A⁻¹B), avoiding post-multiplication.
3. Determinant Zero Geometrical Interpretation: Explaining consistency and geometric meaning when det(A) = 0.