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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▪ genetics

      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.

▪ mathematics

      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 a_rc (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)^r + 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 n × 1 vector and the determinant of A is nonzero, the system of equations AX = B always has a solution.

      For the trivial case of n = 1, the value of the determinant is the value of the single element a₁₁. For n = 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 a_rc is multiplied by the factor (−1)^r + c and by the smaller determinant M_rc 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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