Example 1: Evaluating a 3 × 3 determinant
Evaluate the determinant with rows (1, 2, 3), (0, 1, 4) and (5, 6, 0).
1 × [(1)(0) − (4)(6)] = −24−2 × [(0)(0) − (4)(5)] = 403 × [(0)(6) − (1)(5)] = −15−24 + 40 − 15 = 1
Answer: 1
Determinants of Order 2 and 3 (DSE M2)
A 2 × 2 determinant is ad − bc. A 3 × 3 determinant is expanded along a row: each entry is multiplied by the 2 × 2 determinant left when its row and column are removed, with signs + − +. In M2 determinants are used to find inverse matrices and to solve systems of linear equations.
Determinant Formulas
Order 2
|a b; c d| = ad − bcOrder 3 (first row)
a₁₁(a₂₂a₃₃ − a₂₃a₃₂) − a₁₂(a₂₁a₃₃ − a₂₃a₃₁) + a₁₃(a₂₁a₃₂ − a₂₂a₃₁)Notation
|A| or det ASingular matrix
|A| = 0 ⟺ A has no inverseArea of triangle OAB
½ |a b; c d| for A(a, b), B(c, d) taken anticlockwiseWorked Examples — Determinants
Example 2: Finding k when a determinant is zero
Find the values of k for which the determinant with rows (k, 1, 1), (1, k, 1) and (1, 1, k) equals 0.
k(k² − 1) − 1(k − 1) + 1(1 − k) = k³ − 3k + 2k³ − 3k + 2 = (k − 1)(k² + k − 2) = (k − 1)²(k + 2)(k − 1)²(k + 2) = 0
Answer: k = 1 or k = −2
💡 Exam Traps & Secrets
⚠️ The middle term is subtractedWhen expanding along the first row the signs are
+ − +. Forgetting the minus on the second term is the usual reason a 3 × 3 determinant comes out wrong.⚠️ Show the expansionA calculator gives the value, but the method mark is for the expansion. Write at least one line of working, especially when the entries contain an unknown such as k.
✏️ Self-Test: Quick Interactive Practice
How to study: Click any question below to instantly load it on the coordinate grid and check your steps.
(3 × 3 expansion)Evaluate the determinant with rows (2, −1, 3), (1, 0, 4), (−2, 5, 1).
(Solving |A| = 0)Find k if the determinant with rows (1, 2, 3), (4, k, 6), (7, 8, 9) is zero.
(Area as a determinant)Use a determinant to find the area of triangle OAB with A(4, 1) and B(1, 3).
Frequently Asked Questions
Expand along a row or column. Along the first row: multiply each entry by the 2×2 determinant that remains after deleting its row and column, apply the signs + − +, and add the three results.
The curriculum notes say the properties of determinants are not required. Evaluating by direct expansion is always acceptable.
The matrix is singular: it has no inverse, and a system of equations with that coefficient matrix has either no solution or infinitely many. Geometrically, a 2×2 matrix with zero determinant squashes every shape to zero area.