déterminant

Déterminant is a term used in two distinct domains, French linguistics and mathematics, with related but separate meanings.

In French grammar, déterminant refers to a class of words that accompany nouns to determine their reference. Determinants help specify definiteness, quantity, possession, or interrogative content, and they typically appear before the noun. Subtypes include articles (définis: le, la, les; indéfinis: un, une, des; partitifs: du, de la, des), demonstratives (ce, cet, cette, ces), possessives (mon, ton, son, notre, votre, leur), and determiners expressing quantity or specificity (quelques, plusieurs, tout, chaque, plusieurs, un). Determinants must agree in gender and number with the noun they modify. Examples: le livre, une pomme, du pain, ce livre, mon ami, plusieurs étudiants.

In mathematics, déterminant (determinant in English) denotes a scalar associated with a square matrix that encodes certain essential properties of the linear transformation represented by the matrix. A matrix is invertible if and only if its determinant is nonzero. For a 2×2 matrix [ [a, b], [c, d] ], det = ad − bc; for a 3×3 matrix, det can be computed by the rule of Sarrus or cofactor expansion. The determinant is multilinear and alternating in the columns (or rows), and satisfies det(AB) = det(A) det(B) and det(A^T) = det(A). It also scales by k^n for an n×n matrix when the matrix is multiplied by a scalar k.

Etymology: from Latin determinare, meaning to set boundaries. The two uses share the core idea of defining a reference or boundary—grammatically for nouns, mathematically for linear transformations.