What the determinant is
The determinant is a single number computed from a square matrix. It tells you whether the matrix can be inverted, whether a system of linear equations has a unique solution, and by how much the matrix stretches areas or volumes. This calculator handles 2×2 and 3×3 matrices, the sizes met most often in algebra, geometry and physics courses.
The formulas
- 2×2: det [a b; c d] = ad − bc
- 3×3 (cofactor expansion along the first row): a₁₁(a₂₂a₃₃ − a₂₃a₃₂) − a₁₂(a₂₁a₃₃ − a₂₃a₃₁) + a₁₃(a₂₁a₃₂ − a₂₂a₃₁)
- Trace: the sum of the diagonal entries
- 2×2 inverse: (1 ÷ det) × [d −b; −c a], only when det ≠ 0
The rule of Sarrus
For a 3×3 matrix, write the first two columns again to the right of the matrix. Add the products of the three diagonals going down to the right, then subtract the products of the three diagonals going up to the right. The result is the same as the cofactor expansion; this shortcut only works for 3×3 matrices.
Worked example
Take the matrix with rows (2, 1, 3), (0, −1, 4) and (1, 2, 5).
- Downward diagonals: 2 × (−1) × 5 + 1 × 4 × 1 + 3 × 0 × 2 = −10 + 4 + 0 = −6.
- Upward diagonals: 3 × (−1) × 1 + 1 × 0 × 5 + 2 × 4 × 2 = −3 + 0 + 16 = 13.
- Determinant = −6 − 13 = −19. The matrix is invertible.
- Trace = 2 − 1 + 5 = 6.
For the 2×2 matrix [2 1; 0 −1], the determinant is −2 and the inverse is [0.5 0.5; 0 −1].
What the value means
| Determinant | Meaning |
|---|---|
| not zero | invertible matrix, unique solution to the linear system |
| zero | singular matrix: dependent rows, no inverse |
| absolute value | scale factor for areas (2×2) or volumes (3×3) |
| negative | the transformation flips orientation, like a mirror |
Useful properties
- Swapping two rows changes the sign of the determinant.
- Multiplying a row by k multiplies the determinant by k.
- Adding a multiple of one row to another leaves it unchanged, which is the basis of row reduction.
- The determinant of a product is the product of the determinants: det(AB) = det(A) × det(B).
- A matrix and its transpose have the same determinant.
Where it is used
Determinants appear in Cramer’s rule for solving systems of equations, in finding eigenvalues, in computing the area of a triangle from the coordinates of its vertices, and in 3D graphics, where a negative determinant reveals a mirrored transformation.