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determinant

Also known as matrix determinant, determinant of matrix, determinant of a square matrix

In mathematics, the determinant is a scalar-valued function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the matrix and the linear map represented, on a given basis, by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the corresponding linear map is an isomorphism. However, if the determinant is zero, the matrix is referred to as singular, meaning it does not have an inverse.

AI overview

The determinant is a single number calculated from a square matrix that reveals important properties about that matrix and the mathematical transformation it represents. A nonzero determinant means the matrix can be inverted and represents a reversible transformation, while a zero determinant means the matrix is singular and cannot be inverted.

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Stack Exchange tag: mathoverflow.net/tags/determinants
Commons category: Determinant

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~63 min read

Article

51 sections

Contents

Two by two matrices
First properties
Geometric meaning
Definition
Leibniz formula
3 × 3 matrices
''n'' × ''n'' matrices
Properties
Characterization of the determinant
Immediate consequences
Example
Transpose
Multiplicativity and matrix groups
Laplace expansion
Adjugate matrix
Block matrices
Sylvester's determinant theorem
Sum
Sum identity for 2×2 matrices
Properties of the determinant in relation to other notions
Eigenvalues and characteristic polynomial
Trace
Upper and lower bounds
Derivative
History
Applications
Cramer's rule
Linear independence
Cross Product
Orientation of a basis
Volume and Jacobian determinant
Areas and Collinearity
Abstract algebraic aspects {{anchor|Abstract formulation}}
Determinant of an endomorphism
Square matrices over commutative rings
Exterior algebra
Berezin integral
Generalizations and related notions
Determinants for finite-dimensional algebras
Infinite matrices
Operators in von Neumann algebras
Related notions for non-commutative rings
Calculation
Gaussian elimination
Decomposition methods
Further methods
See also
Notes
References
Historical references
External links

The determinant is completely determined by the two following properties: the determinant of a product of matrices is the product of their determinants, and the determinant of a triangular matrix is the product of its diagonal entries.