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In linear algebra, the determinant is a function which associates a number to every square matrix. For instance, the 2by2 matrix
The determinant of A is also sometimes denoted by A, but this notation should be avoided as it is also used to denote other matrix functions, such as the square root of AA^{*}.

Historically, determinants were considered before matrices. Originally, a determinant was defined as a property of a system of linear equations. The determinant "determines" whether the system has a unique solution (which occurs precisely if the determinant is nonzero). In this sense, twobytwo determinants were considered by Cardano at the end of the 16th century and larger ones by Leibniz about 100 years later.
Determinants are used to characterize invertible matrices, and to explicitly describe the solution to a system of linear equations with Cramer's rule. It can be used to find the eigenvalues of the matrix A through the characteristic polynomial p(x) = det(AxI_{n}).
One often thinks of the determinant as assigning a number to every sequence of n vectors in R^{n}, by using the square matrix whose columns are the given vectors. With this understanding, the sign of the determinant of a basis can be used to define the notion of orientation[?] in Euclidean spaces.
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 : R^{n} > R^{n} is represented by the matrix A, and S is any measurable subset of R^{n}, then the volume of f(S) is given by det(A) × volume(S). More generally, if the linear map f : R^{n} > R^{m} is represented by the mbyn matrix A, and S is any measurable subset of R^{n}, then the ndimensional volume of f(S) is given by √(det(A^{T}A)) × volume(S).
Suppose A = (A_{i,j}) is a square matrix.
If A is a 1by1 matrix, then det(A) = A_{1,1}. If A is a 2by2 matrix, then det(A) = A_{1,1} · A_{2,2}  A_{2,1} · A_{1,2}. For a 3by3 matrix A, the formula is more complicated:
For a general nbyn matrix, the determinant was defined by Gottfried Leibniz with what is now known as the Leibniz formula:
This formula contains n! summands and is therefore impractical to use if n is bigger than 3.
In general, determinants can be computed with the Gauss algorithm using the following rules:
Explicitly, starting out with some matrix, use the last three rules to convert it into a triangular matrix, then use the first rule to compute its determinant.
It is also possible to expand a determinant along a row or column using Laplace's formula. To do this along row i, say, we write
where A(ij) denotes the n1 by n1 matrix resulting from A by removing the ith row and jth column.
The determinant is a multiplicative map in the sense that
It is easy to see that det(rI_{n})=r^{n} and thus
If A is invertible, then
A matrix and its transpose have the same determinant:
If A and B are similar, i.e. if there exists an invertible matrix X such that A = X^{1}BX, then by the multiplicative property,
There exist matrices which have the same determinant but are not similar.
If A is a square nbyn matrix with real or complex entries and if λ_{1},...,λ_{n} are the (complex) eigenvalues of A listed according to their algebraic multiplicities, then
From the connection between the determinant and the eigenvalues, one can derive a connection between the trace function, the exponential function, and the determinant:
The determinant of real square matrices is a polynomial function from R^{n×n} to R, and as such is everywhere differentiable. Its derivative can be expressed using Jacobi's formula:
It makes sense to define the determinant for matrices whose entries come from any commutative ring. The computation rules, the Leibniz formula and the compatibility with matrix multiplication remain valid, except that now a matrix A is invertible if and only if det(A) is an invertible element of the ground ring.
Abstractly, one may define the determinant as a certain antisymmetric multilinear map as follows: if R is a commutative ring and M = R^{n} denotes the free Rmodule[?] with n generators, then
Linear Algebraists prefer to use the multilinear map approach to define determinant, whereas algebraists prefer to use the Leibniz formula.
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