determinant
Sign in to saveIn 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.
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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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Article
51 sectionsContents
- 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- Historical references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
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.
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.