In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the determinant is a
scalar value that is a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
of the entries of a
square matrix. It characterizes some properties of the matrix and the
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
represented by the matrix. In particular, the determinant is nonzero if and only if the matrix is
invertible
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
and the linear map represented by the matrix is an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
. 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 is denoted , , or .
The determinant of a matrix is
:
and the determinant of a matrix is
:
The determinant of a 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 different entries, and the number of these summands is
the
factorial of (the product of the first positive integers). The
Laplace expansion
In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression of the determinant of an matrix as a weighted sum of minors, which are the determinants of some submatrices of . Spec ...
expresses the determinant of a matrix as a
linear combination of determinants of
submatrices.
Gaussian elimination express the determinant as the product of the diagonal entries of a
diagonal matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal m ...
that is obtained by a succession of
elementary row operation In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation. The elementary matrices generate the general linear group GL''n''(F) when F is a field. Left multiplication (pre-multipl ...
s.
Determinants can also be defined by some of their properties: the determinant is the unique function defined on the matrices that has the four following properties. The determinant of the
identity matrix is ; the exchange of two rows (or of two columns) multiplies the determinant by ; 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
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s. In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the signed -dimensional
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...
of a -dimensional
parallelepiped is expressed by a determinant. This is used in
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
with
exterior differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications ...
s and the
Jacobian determinant
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables ...
, in particular for
changes of variables in
multiple integral
In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, or . Integrals of a function of two variables over a region in \mathbb^2 (the real-number ...
s.
2 × 2 matrices
The determinant of a matrix
is denoted either by "" or by vertical bars around the matrix, and is defined as
:
For example,
:
First properties
The determinant has several key properties that can be proved by direct evaluation of the definition for
-matrices, and that continue to hold for determinants of larger matrices. They are as follows: first, the determinant of the
identity matrix is 1.
Second, the determinant is zero if two rows are the same:
:
This holds similarly if the two columns are the same. Moreover,
:
Finally, if any column is multiplied by some number
(i.e., all entries in that column are multiplied by that number), the determinant is also multiplied by that number:
:
Geometric meaning
If the matrix entries are real numbers, the matrix can be used to represent two
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
s: one that maps the
standard basis vectors to the rows of , and one that maps them to the columns of . In either case, the images of the basis vectors form a
parallelogram that represents the image of the
unit square
In mathematics, a unit square is a square whose sides have length . Often, ''the'' unit square refers specifically to the square in the Cartesian plane with corners at the four points ), , , and .
Cartesian coordinates
In a Cartesian coordin ...
under the mapping. The parallelogram defined by the rows of the above matrix is the one with vertices at , , , and , as shown in the accompanying diagram.
The absolute value of is the area of the parallelogram, and thus represents the scale factor by which areas are transformed by . (The parallelogram formed by the columns of is in general a different parallelogram, but since the determinant is symmetric with respect to rows and columns, the area will be the same.)
The absolute value of the determinant together with the sign becomes the ''oriented area'' of the parallelogram. The oriented area is the same as the usual
area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape
A shape or figure is a graphics, graphical representation of an obje ...
, except that it is negative when the angle from the first to the second vector defining the parallelogram turns in a clockwise direction (which is opposite to the direction one would get for the
identity matrix).
To show that is the signed area, one may consider a matrix containing two vectors and representing the parallelogram's sides. The signed area can be expressed as for the angle ''θ'' between the vectors, which is simply base times height, the length of one vector times the perpendicular component of the other. Due to the
sine this already is the signed area, yet it may be expressed more conveniently using the
cosine of the complementary angle to a perpendicular vector, e.g. , so that , which can be determined by the pattern of the
scalar product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
to be equal to :
:
Thus the determinant gives the scaling factor and the orientation induced by the mapping represented by ''A''. When the determinant is equal to one, the linear mapping defined by the matrix is
equi-areal and orientation-preserving.
The object known as the ''
bivector In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. If a scalar is considered a degree-zero quantity, and a vector is a degree-one quantity, then a bivector ca ...
'' is related to these ideas. In 2D, it can be interpreted as an ''oriented plane segment'' formed by imagining two vectors each with origin , and coordinates and . The bivector magnitude (denoted by ) is the ''signed area'', which is also the determinant .
If an
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
matrix ''A'' is written in terms of its column vectors