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Determinant
In mathematics, the determinant is a scalar-valued function of the entries of a square matrix that has many properties which make it fundamental for the study of square matrices and linear transformations represented by them. The determinant of a matrix A is commonly denoted det(A), det A, or |A|. For n by n matrices with n ≤ 3 {\displaystyle n\leq 3} , the determinant is explicitly described as det [ a ] = a ( n = 1 ) , det [ a b c d ] = a d − b c ( n = 2 ) , det [ a b c d e f g h i ] = a e i + b f g + c d h − c e g − b d i − a f h ( n = 3 ) . {\displaystyle {\begin{aligned}\det[a]&=a\qquad (n=1),\\[10pt]\det {\begin{bmatrix}a&b\\c&d\end{bmatrix}}&=ad-bc\qquad (n=2),\\[10pt]\det {\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}}&=aei+bfg+cdh-ceg-bdi-afh\qquad (n=3).\end{aligned}}} There are several equivalent ways to define the determinant of an n × n {\displaystyle n\times n} matrix. A simple and conceptual one is that it is the unique function that maps a product of matrices to a product of scalars and whose value on a triangular matrix is the product of the diagonal entries. So, the determinant of the product of two square matrices is the product of their determinant. A square matrix is invertible if and only if its determinant has a multiplicative inverse. In the common case where the entries belong to a field, a matrix is invertible if and only if the determinant is nonzero. The adjugate matrix provides an explicit expression of the inverse matrix. A consequence of the multiplicative property is that two similar matrices have the same determinant, and therefore the determinant of a linear transformation is well defined. For real linear transformations, the value of the determinant is the scale factor by which the transformation alters every volume in the space (positive if the orientation is preserved, negative if the orientation is reversed, zero if the transformation is not bijective). The determinant is linear with respect to each row and each column of the matrix and is therefore a multilinear function. This means that it is a linear combination of the entries of each row or column of the matrix. Laplace expansion expresses the coefficients of this linear combination as determinants (of one dimension less), called cofactors. Laplace expansion is useful for computing determinants when the rows or columns are very sparse, typically when there are only one or two nonzero entries in a row or column. Determinants can be efficiently computed with Gaussian elimination and singular value decomposition. Determinants can also be theoretically computed by means of the Leibniz formula or Laplace expansion, but this is inefficient, except in special cases. Determinants occur throughout mathematics. Cramer's rule expresses the solution of a system of linear equations in terms of determinants. The characteristic polynomial of a square matrix, whose roots are the eigenvalues, is defined as a determinant with polynomial entries. In geometry, determinants are used to define or compute areas and volumes, as well as space orientation and colinearity. In analysis, the Jacobian determinant is used for change of variables in multiple integrals and its non-nullity is one of the hypotheses of the implicit function theorem in several variables. In algebraic geometry, the Hessian determinant allows finding inflexion points of an algebraic plane curve.