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# determinant

/di terr"meuh neuhnt/, n.
1. a determining agent or factor.
2. Math. an algebraic expression of the sum of products of elements, each with an appropriate algebraic sign, usually written in a square array and used in the solution of systems of linear equations.
3. Also called antigenic determinant, epitope. Immunol. any site on an antigen molecule at which an antibody can bind, the chemical structure of the site determining the specific combining antibody.
4. Genetics Archaic. a gene.
[1600-10; < L determinant- (s. of determinans), prp. of determinare. See DETERMINE, -ANT]

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In linear algebra, a numerical value associated with a matrix having the same number of rows as columns.

It is particularly useful in solving systems of (linear) equations and in the study of vectors. For a two-by-two matrix, the determinant is the product of the upper left and lower right terms minus the product of the lower left and upper right terms. Determinants of larger matrices involve more complicated arithmetic combinations of the terms and are usually solved using a calculator or computer.

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in genetics, the term used in the late 19th century by the German biologist August Weismann to describe the component of hereditary material, or germ plasm, that specifies the characteristics of different cells.

in linear and multilinear algebra, a value, denoted det A, associated with a square matrix A of n rows and n columns. Designating any element of the matrix by the symbol arc (the subscript r identifies the row and c the column), the determinant is evaluated by finding the sum of n! terms, each of which is the product of the coefficient (−1)c and n elements, no two from the same row or column. Determinants are of use in ascertaining whether a system of n equations in n unknowns has a solution. If B is an × 1 vector and the determinant of A is nonzero, the system of equations AX = B always has a solution.

For the trivial case of = 1, the value of the determinant is the value of the single element a11. For = 2, the matrix is

and the determinant is a11a22 −  a12a21.

Larger determinants ordinarily are evaluated by a stepwise process, expanding them into sums of terms, each the product of a coefficient and a smaller determinant. Any row or column of the matrix is selected, each of its elements arc is multiplied by the factor (−1)r + c and by the smaller determinant Mrc formed by deleting the rth row and cth column from the original array. Each of these products is expanded in the same way until the small determinants can be evaluated by inspection. At each stage, the process is facilitated by choosing the row or column containing the most zeros.

For example, the determinant of the matrix

is most easily evaluated with respect to the second column:

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Universalium. 2010.

### Look at other dictionaries:

• déterminant — déterminant, ante [ detɛrminɑ̃, ɑ̃t ] adj. et n. m. • av. 1662; de déterminer I ♦ Adj. 1 ♦ Qui détermine (1o); qui sert à déterminer. ⇒ caractéristique. Motif déterminant. Cause déterminante. 2 ♦ Qui décide d une chose ou d une action. ⇒… …   Encyclopédie Universelle

• determinant — DETERMINÁNT, Ă, determinanţi, te, adj., s.m. I. adj. 1. Care determină sau este de natură să determine ceva; hotărâtor. 2. (Despre cuvinte sau propoziţii; adesea substantivat) Care precizează sensul altui cuvânt sau al altei propoziţii cu care… …   Dicționar Român

• Determinant — Déterminant Cette page d’homonymie répertorie les différents sujets et articles partageant un même nom …   Wikipédia en Français

• déterminant — déterminant, ante (dé tèr mi nan, nan t ) adj. Qui détermine. Ce motif est déterminant. •   J expose celles de mes raisons que je pouvais dire sans compromettre Mme Levasseur et sa famille ; car les plus déterminantes venaient de là, et je les… …   Dictionnaire de la Langue Française d'Émile Littré

• Determinant — De*ter mi*nant, n. 1. That which serves to determine; that which causes determination. [1913 Webster] 2. (Math.) The sum of a series of products of several numbers, these products being formed according to certain specified laws; Note: thus, the… …   The Collaborative International Dictionary of English

• déterminant — DÉTERMINANT, ANTE. adject. Qui détermine, qui sert à déterminer. Ce motif est déterminant. C est une raison déterminante …   Dictionnaire de l'Académie Française 1798

• Determinant — De*ter mi*nant, a. [L. determinans, p. pr. of determinare: cf. F. d[ e]terminant.] Serving to determine or limit; determinative. [1913 Webster] …   The Collaborative International Dictionary of English

• determinant — I noun agent, authority, background, base, basis, consideration, constitutive element, contributing force, contributor, decisive factor, determining circumstance, determining element, determining influence, driving force, element, factor,… …   Law dictionary

• determinant — c.1600 (adj.); 1680s (n.), from L. determinantem (nom. determinans), prp. of determinare (see DETERMINE (Cf. determine)) …   Etymology dictionary

• determinant — determìnant m <G mn nātā> DEFINICIJA gram. oznaka, sastavnica imenske sintagme koja se odnosi na imenicu, o njoj ovisi, tj. pobliže je označuje (kao vrsta riječi to su posebice član, pridjev, posesiv, demonstrativ, broj) ETIMOLOGIJA vidi… …   Hrvatski jezični portal

• determinant — antecedent, *cause, reason, occasion Analogous words: factor (see ELEMENT): *influence, weight, authority …   New Dictionary of Synonyms

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