What it is
The determinant is a number assigned to a square n×n matrix, written det A or |A|. It shows how the linear transformation given by the matrix changes areas (in 2D) and volumes (in 3D), and immediately answers the key question: is the matrix invertible and does the system of equations have a unique solution.
How it is calculated
- 2×2:
det {{a, b}, {c, d}} = a·d − b·c. - 3×3: expansion along the first row:
a11(a22a33 − a23a32) − a12(a21a33 − a23a31) + a13(a21a32 − a22a31)(the rule of Sarrus gives the same). - n×n: cofactor expansion along a row (minors with alternating signs) or Gaussian elimination to triangular form — then the determinant is the product of the diagonal.
Example
| A | ||
|---|---|---|
| 2 | 1 | 3 |
| 0 | 4 | 1 |
| 5 | 2 | 0 |
- 2·(4·0 − 1·2) = 2·(−2) = −4
- −1·(0·0 − 1·5) = −1·(−5) = +5
- +3·(0·2 − 4·5) = 3·(−20) = −60
- det A = −4 + 5 − 60 = −59
The matrix {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}} has determinant 0: the third row is a combination of the first two (2·row2 − row1), so the matrix is singular.
Properties
- det A = 0 ⇔ the matrix is singular: it has no inverse, its rows (columns) are linearly dependent, and A·x = b has no unique solution.
- det(A·B) = det A · det B; det AT = det A; det A−1 = 1 / det A.
- Swapping two rows flips the sign; a common factor of a row can be taken outside; adding a multiple of one row to another does not change the determinant.
- det(k·A) = kn·det A for an n×n matrix.
- |det A| is the factor by which the transformation scales area or volume; a negative sign means a mirror reflection (change of orientation).
Where it is used
- Systems of linear equations, Cramer's rule. For 3x + y = 9, 2x + 4y = 16: Δ = 3·4 − 1·2 = 10, Δx = 9·4 − 1·16 = 20, Δy = 3·16 − 9·2 = 30, so x = 20/10 = 2, y = 30/10 = 3 (check: 3·2 + 3 = 9, 2·2 + 4·3 = 16).
- Invertibility check — before computing an inverse or solving a system, make sure det ≠ 0.
- Geometry — area of a parallelogram or triangle from vertex coordinates, volume of a parallelepiped (scalar triple product), checking whether three points are collinear (det = 0).
- 3D graphics and physics — a negative determinant of a transform means the model is mirrored; the Jacobian when changing variables in integrals.
- Eigenvalues are found from det(A − λE) = 0 — the basis of stability and vibration analysis and principal component analysis.
- Statistics and ML — the determinant of the covariance matrix appears in the multivariate normal distribution.
- Economics — solvability of the Leontief model: the matrix (E − A) must be non-singular.
How to use
- Set the size with the add/remove row buttons — the matrix always stays square, columns follow automatically.
- Fill in the cells (fractions and expressions work) or enter the matrix as a string
{n,n,elements…}, e.g.{3,3,2,1,3,0,4,1,5,2,0}. - The determinant is recalculated automatically and shown in the det field.