determinante

Determinante, known in English as the determinant, is a scalar associated with a square matrix that describes how the linear transformation represented by the matrix changes volumes and orientation. It is a function det: F^n×n → F that is linear in each row (multilinear) and changes sign if two rows are swapped (alternating), with det(I) = 1. The determinant is zero precisely when the rows (or columns) are linearly dependent.

For a 2×2 matrix [a b; c d], det = ad − bc. In general, det A can be defined by a sum over permutations (the Leibniz formula) or computed efficiently via row reduction or LU decomposition. A triangular matrix has determinant equal to the product of its diagonal entries.

Key properties include det(AB) = det(A) det(B); det(A^T) = det(A); det(-A) = (−1)^n det(A). A matrix is invertible exactly when det(A) ≠ 0, and det(A) varies continuously with the entries of A. The determinant also equals the volume scaling factor of the linear map associated with A, and its sign reflects orientation.

Applications include solving linear systems (Cramer's rule), testing invertibility, and studying eigenvalues through the characteristic polynomial det(A − λI). Geometrically, determinants relate to areas and volumes of parallelograms and parallelpipeds spanned by rows or columns.

Notation commonly uses det(A) or |A|. The determinant has roots in 17th- to 18th-century work of mathematicians such as Leibniz, Cramer, and Gauss, and it became a central tool in linear algebra and analytic geometry.

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