Matrix Determinant Calculator
Compute the determinant of any square matrix (2×2 to 5×5) with step-by-step working. Uses cofactor expansion for 2×2/3×3 and LU decomposition for larger matrices.
| Curriculum | Coverage | Method taught |
|---|---|---|
| 🇬🇧 UK A-Level Further Maths | 2×2 and 3×3 | Cofactor expansion, Sarrus rule |
| 🌐 IB HL Mathematics | 2×2 and 3×3 | Cofactor expansion (GDC allowed) |
| 🇩🇪 Germany Abitur | Up to 3×3 | Sarrus rule (preferred in Germany) |
| 🇺🇸 US Linear Algebra | n×n general | LU decomposition, row reduction |
| 🇯🇵 Japan (数学C) | 2×2 and 3×3 | Cofactor expansion |
| 💻 ML / Numerical Computing | n×n | LU decomposition (numerically stable) |
Properties of Determinants
- det(AB) = det(A) × det(B) — product rule
- det(Aᵀ) = det(A) — transpose has same determinant
- det(A⁻¹) = 1 / det(A) — inverse determinant
- det(kA) = kⁿ det(A) for n×n matrix A scaled by k
- Swapping two rows negates the determinant
- Adding a multiple of one row to another leaves the determinant unchanged
Frequently Asked Questions
What is the Sarrus rule and when is it used?
The Sarrus rule is a mnemonic for computing 3×3 determinants: write the matrix, repeat the first two columns to the right, then sum the three main diagonals (going down-right) and subtract the three anti-diagonals (going down-left). It is only valid for 3×3 and is popular in German, French, and Spanish high school curricula. It does NOT generalize to 4×4 or larger matrices — a common mistake is to attempt the "Sarrus method" on 4×4 matrices.
How is the determinant used in Cramer's rule?
Cramer's rule solves a linear system Ax = b by computing xᵢ = det(Aᵢ) / det(A), where Aᵢ is A with column i replaced by b. It requires det(A) ≠ 0 (non-singular) and is computationally expensive for large systems (O(n!) operations), so it's mainly used for 2×2 and 3×3 systems in exams. Gaussian elimination is preferred computationally.