determinantdɪˈtɜr mə nənt
determinant (n)
- plural
- determinants
English Definitions:
determinant, determiner, determinative, determining factor, causal factor (noun)
a determining or causal element or factor
"education is an important determinant of one's outlook on life"
antigenic determinant, determinant, epitope (noun)
the site on the surface of an antigen molecule to which an antibody attaches itself
determinant (adj)
a square matrix used to solve simultaneous equations
deciding(a), determinant, determinative, determining(a) (adj)
having the power or quality of deciding
"the crucial experiment"; "cast the deciding vote"; "the determinative (or determinant) battle"
determinant (Noun)
A determining factor; an element that determines the nature of something
determinant (Noun)
The unique scalar function over square matrices which is distributive over matrix multiplication, multilinear in the rows and columns, and takes the value of 1 for the unit matrix. Abbreviation: det
determinant (Noun)
A substance that causes a cell to adopt a particular fate.
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well. The determinant provides important information when the matrix is that of the coefficients of a system of linear equations, or when it corresponds to a linear transformation of a vector space: in the first case the system has a unique solution exactly when the determinant is nonzero; when the determinant is zero there are either no solutions or many solutions. In the second case that same condition means that the transformation has an inverse operation. A geometric interpretation can be given to the value of the determinant of a square matrix with real entries: the absolute value of the determinant gives the scale factor by which area or volume is multiplied under the associated linear transformation, while its sign indicates whether the transformation preserves orientation. Thus a 2 × 2 matrix with determinant −2, when applied to a region of the plane with finite area, will transform that region into one with twice the area, while reversing its orientation.
Determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the linear map represented by the matrix is an isomorphism. The determinant of a product of matrices is the product of their determinants (the preceding property is a corollary of this one). The determinant of a matrix A is denoted det(A), det A, or |A|. The determinant of a 2 × 2 matrix is | a b c d | = a d − b c , {\displaystyle {\begin{vmatrix}a&b\\c&d\end{vmatrix}}=ad-bc,} and the determinant of a 3 × 3 matrix is | a b c d e f g h i | = a e i + b f g + c d h − c e g − b d i − a f h . {\displaystyle {\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}=aei+bfg+cdh-ceg-bdi-afh.} The determinant of a n × n matrix can be defined in several equivalent ways. Leibniz formula expresses the determinant as a sum of signed products of matrix entries such that each summand is the product of n different entries, and the number of these summands is n ! , {\displaystyle n!,} the factorial of n (the product of the n first positive integers). The Laplace expansion expresses the determinant of a n × n matrix as a linear combination of determinants of ( n − 1 ) × ( n − 1 ) {\displaystyle (n-1)\times (n-1)} submatrices. Gaussian elimination express the determinant as the product of the diagonal entries of a diagonal matrix that is obtained by a succession of elementary row operations. Determinants can also be defined by some of their properties: the determinant is the unique function defined on the n × n matrices that has the four following properties. The determinant of the identity matrix is 1; the exchange of two rows (or of two columns) multiplies the determinant by −1; multiplying a row (or a column) by a number multiplies the determinant by this number; and adding to a row (or a column) a multiple of another row (or column) does not change the determinant. Determinants occur throughout mathematics. For example, a matrix is often used to represent the coefficients in a system of linear equations, and determinants can be used to solve these equations (Cramer's rule), although other methods of solution are computationally much more efficient. Determinants are used for defining the characteristic polynomial of a matrix, whose roots are the eigenvalues. In geometry, the signed n-dimensional volume of a n-dimensional parallelepiped is expressed by a determinant. This is used in calculus with exterior differential forms and the Jacobian determinant, in particular for changes of variables in multiple integrals.
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