Determinant algebra, a determinant is a function depending on n that associates a scalar, det(A), to an n×n square matrix A. The fundamental geometric meaning of a determinant is a scale factor for measure 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 nonnegative integer n, there is a unique determinant function for the n×n matrices over any commutative ring R.