deta

The determinant of a square matrix A, denoted det(A) or |A|, is a scalar function that assigns to A a single number. For an n×n matrix, det(A) encodes how the linear transformation x ↦ Ax scales volumes and whether it preserves or reverses orientation. A is invertible if and only if det(A) ≠ 0; equivalently, det(A) = 0 signals that A is singular.

Definition and key properties: det is a polynomial in the entries of A, and it is alternating with respect to the rows (or columns). Fundamental properties include det(I) = 1, det(AB) = det(A) det(B), det(A^T) = det(A), and det(kA) = k^n det(A) for a scalar k and n×n A. If det(A) ≠ 0, then A has an inverse and det(A^{-1}) = 1/det(A). The determinant is continuous in the matrix entries and satisfies det(A) = 0 if and only if the rows (or columns) are linearly dependent.

Computation: determinants can be computed by expansion by minors, or more practically by row reduction to an upper triangular form. If the reduction uses row interchanges, det(A) changes sign with each swap; if Q is obtained from A by row operations to U, then det(A) = det(Q) det(U), and det(U) is the product of its diagonal entries.

Formulas for small sizes: for a 2×2 matrix [ [a, b], [c, d] ], det = ad − bc. for a 3×3 matrix [ [a, b, c], [d, e, f], [g, h, i] ], det = a(ei − fh) − b(di − fg) + c(dh − eg).

Geometric and applications: |det(A)| equals the oriented volume scaling factor of the linear map; its sign indicates orientation. Determinants are used to test linear independence, solve linear systems (e.g., Cramer’s rule in appropriate cases), change of variables in integrals, and to relate to eigenvalues (det(A) is the product of eigenvalues). Determinants are defined only for square matrices.