Determinant

In linear algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. Determinants are important both in calculus, where they enter the substitution rule for several variables, and in multilinear algebra.

Applications

Determinants are used to characterize invertible matrices (namely as those matrices, and only those matrices, with non-zero determinants), and to explicitly describe the solution to a system of linear equations with Cramer's rule. They can be used to find the eigenvalues of the matrix A through the characteristic polynomial p(x) = det(xI - A) (where I is the identity matrix of the same format as A).

Related Topics:
Invertible matrices - Linear equation - Cramer's rule - Eigenvalue - Characteristic polynomial - Identity matrix

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One often thinks of the determinant as assigning a number to every sequence of n vectors in Bbb{R}^n, by using the square matrix whose columns are the given vectors.

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With this understanding, the sign of the determinant of a basis can be used to define the notion of orientation in Euclidean spaces. The determinant of a set of vectors is positive if the vectors form a right-handed coordinate system, and negative if left-handed.

Related Topics:
Basis - Orientation - Euclidean space - Positive - Coordinate system

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Determinants are used to calculate volumes in vector calculus: the absolute value of the determinant of real vectors is equal to the volume of the parallelepiped spanned by those vectors. As a consequence, if the linear map f: Bbb{R}^n ightarrow Bbb{R}^n is represented by the matrix A, and S is any measurable subset of Bbb{R}^n, then the volume of f(S) is given by left| det(A) ight| imes operatorname{volume}(S). More generally, if the linear map f: Bbb{R}^n ightarrow Bbb{R}^m is represented by the m-by-n matrix A, and S is any measurable subset of Bbb{R}^{n}, then the n-dimensional volume of f(S) is given by sqrt{det(A^ op A)} imes operatorname{volume}(S). By calculating the volume of the tetrahedron bounded by four points, they can be used to identify skew lines.

Related Topics:
Volume - Vector calculus - Absolute value - Parallelepiped - Linear map - Measurable - Subset - Dimension - Tetrahedron - Skew line

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The volume of any tetrahedron, given its vertices a, b, c, and d, is (1/6)·|det(a−b, b−c, c−d)|, or any other combination of pairs of vertices that form a simply connected graph.

Related Topics:
Tetrahedron - Det - Graph

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