Editing Determinant (section)

{{Short description|In mathematics, invariant of square matrices}}
{{about|mathematics|determinants in epidemiology|Risk factor|determinants in immunology|Epitope}}

In [[mathematics]], the '''determinant''' is a [[Scalar (mathematics)|scalar]]-valued [[function (mathematics)|function]] of the entries of a [[square matrix]]. The determinant of a matrix {{math|''A''}} is commonly denoted {{math|det(''A'')}}, {{math|det ''A''}}, or {{math|{{abs|''A''}}}}. Its value characterizes some properties of the matrix and the [[linear map]] represented, on a given [[basis (linear algebra)|basis]], by the matrix. In particular, the determinant is nonzero [[if and only if]] the matrix is [[invertible matrix|invertible]] and the corresponding linear map is an [[linear isomorphism|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.

The determinant of a {{math|2 × 2}} matrix is
:<math>\begin{vmatrix} a & b\\c & d \end{vmatrix}=ad-bc,</math>
and the determinant of a {{math|3 × 3}} matrix is
:<math> \begin{vmatrix} 
a & b & c \\ 
d & e & f \\ 
g & h & i 
\end{vmatrix} = aei + bfg + cdh - ceg - bdi - afh.</math>

The determinant of an {{math|''n'' × ''n''}} matrix can be defined in several equivalent ways, the most common being [[Leibniz formula for determinants|Leibniz formula]], which expresses the determinant as a sum of <math>n!</math> (the [[factorial]] of {{mvar|n}}) signed products of matrix entries. It can be computed by the [[Laplace expansion]], which expresses the determinant as a [[linear combination]] of determinants of submatrices, or with [[Gaussian elimination]], which allows computing a [[row echelon form]] with the same determinant, equal to the product of the diagonal entries of the row echelon form.

Determinants can also be defined by some of their properties. Namely, the determinant is the unique function defined on the {{math|''n'' × ''n''}} matrices that has the four following properties: 
# The determinant of the [[identity matrix]] is {{math|1}}.
# The exchange of two rows multiplies the determinant by {{math|−1}}.
# Multiplying a row by a number multiplies the determinant by this number. 
# Adding a multiple of one row to another row does not change the determinant. 

The above properties relating to rows (properties 2–4) may be replaced by the corresponding statements with respect to columns.

The determinant is invariant under [[matrix similarity]]. This implies that, given a linear [[endomorphism]] of a [[finite-dimensional vector space]], the determinant of the matrix that represents it on a [[basis (vector space)|basis]] does not depend on the chosen basis. This allows defining the ''determinant'' of a linear endomorphism, which does not depend on the choice of a [[coordinate system]].

Determinants occur throughout mathematics. For example, a matrix is often used to represent the [[coefficient]]s 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 square matrix, whose roots are the [[eigenvalue]]s. In [[geometry]], the signed {{mvar|n}}-dimensional [[volume]] of a {{mvar|n}}-dimensional [[parallelepiped]] is expressed by a determinant, and the determinant of a linear [[endomorphism]] determines how the [[orientability|orientation]] and the {{mvar|n}}-dimensional volume are transformed under the endomorphism. This is used in [[calculus]] with [[exterior differential form]]s and the [[Jacobian determinant]], in particular for [[Integration by substitution#Substitution for multiple variables|changes of variables]] in [[multiple integral]]s.