Matrix Determinant 2×2 / 3×3

Determinant: det(A) for 2×2, 3×3, and n×n

The determinant is a scalar that summarises key properties of a square matrix. For a 2×2 matrix [[a, b], [c, d]], det = a·d - b·c. For 3×3 you can use the Sarrus rule (sum of three left-to-right diagonals minus three right-to-left diagonals) or cofactor expansion. For n×n, the standard methods are Laplace expansion along a row/column or Gaussian elimination (faster, O(n³)). Example: det([[2, 3], [1, 4]]) = 2·4 - 3·1 = 5. Key properties: det ≠ 0 means the matrix is invertible and the associated linear system Ax = b has a unique solution; det = 0 means the matrix is singular and the system has either no solution or infinitely many. Geometrically, |det| equals the volume of the parallelepiped formed by the row (or column) vectors — area in 2D, volume in 3D — and the sign indicates orientation (positive = preserves orientation, negative = reflection). Cramer's rule gives xᵢ = det(Aᵢ) / det(A) for solving linear systems.

Applications

Solving linear systems (Cramer's rule), change of variables in multiple integrals (the Jacobian determinant in 2D/3D), 2D/3D linear transformations (scaling, rotation, shear — the determinant tells you how areas/volumes scale), polygon area via the shoelace formula, computer graphics, and machine learning (PCA uses eigenvalues; the determinant of the covariance matrix measures total variance and appears in the multivariate Gaussian density).

FAQ

Why does det = 0 mean no inverse? Because the inverse formula is A⁻¹ = (1/det) · adj(A); dividing by 0 is impossible. Geometrically, det = 0 means the matrix collapses space to a lower dimension (a plane onto a line, for example).

Does det(AB) = det(A)·det(B)? Yes — the determinant is multiplicative for square matrices of the same size. It follows that det(A⁻¹) = 1/det(A).

Is there a determinant for non-square matrices? No. The determinant is only defined for square matrices. For rectangular ones, related concepts are rank, singular values, and the Gram determinant det(AᵀA).