Determinants
Determinants questions cover expansion, the effect of row and column operations, area of a triangle, adjoint and inverse, and solving a system of linear equations by the matrix method. A typical paper has a 1-mark value, a 3-mark property-based evaluation and a 4–6 mark system-solving problem.
Properties that save time
- Swapping two rows changes the sign of the determinant.
- If two rows or columns are identical, the determinant is zero.
- Multiplying one row by k multiplies the determinant by k; for an n×n matrix, |kA| = kⁿ|A|.
- Adding a multiple of one row to another leaves the value unchanged.
- |adj A| = |A|^(n−1) and adj A = |A|·A⁻¹.
- Area of a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) is ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|.
Worked example
Solve x + 2y = 4, 3x + 5y = 7. Here A = [[1, 2], [3, 5]], |A| = 5 − 6 = −1, so A⁻¹ = [[−5, 2], [3, −1]]. Then X = A⁻¹B = [[−5, 2], [3, −1]]·[4, 7]ᵗ = [−6, 5]ᵗ, so x = −6, y = 5. Substitution checks both equations.
Traps
- Writing |kA| = k|A| for a 3×3 matrix.
- Forgetting the absolute value in the triangle-area formula.
- Declaring a system inconsistent without checking |A| = 0 separately.