Det

Det is a term with several distinct meanings in different domains. In Scandinavian languages such as Danish, Norwegian Bokmål, and Swedish, det is a neuter pronoun meaning “it” or “that.” It is used in both generic and specific references and features prominently in idiomatic and syntactic constructions, including impersonal expressions like “Det regner” (It is raining) or “Det finns” (There is/are). It can also function in phrases such as “det här” (this) and “det där” (that).

In mathematics, det denotes the determinant of a square matrix. The determinant is a scalar value that encodes important properties of a linear transformation, including volume scaling and invertibility. A matrix A is invertible if and only if det(A) ≠ 0. The determinant is multiplicative, meaning det(AB) = det(A) det(B), and it remains unchanged under transposition: det(A^T) = det(A). For a diagonal matrix, det is simply the product of the diagonal entries; in general, determinants can be computed via row reduction, cofactor expansion, or eigenvalue products (det(A) equals the product of A’s eigenvalues, counting multiplicities). In calculus, the determinant appears in change-of-variables formulas, where the Jacobian determinant scales volumes under a differentiable map.

Across fields, det is typically written as det(A) or |A| to denote the determinant of matrix A. Its use as a symbol and concept is central in linear algebra, multivariable calculus, and many applied disciplines.