In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. Determinants are important both in calculus, where they enter the substitution rule for several variables, and in multilinear algebra.
For a fixed positive integer n, there is a unique determinant function for the n×n matrices over any commutative ring R. In particular, this function exists when R is the field of real or complex numbers.