a mathematical structure that is used in linear algebra and is useful for solving linear systems and for computing the inverse of matrices

a number determined by the sum of a series of multiplications of the elements in the matrix selected so that a number in each row and column occurs once in each product

a polynomial of the elements of a square matrix

a value computed for a square matrix (if I remember right)

a value representing sums and products of a square matrix

A real number represented by a square array of numbers.

A quantity of a matrix that characterizes the amount of expansion or contraction that the matrix inflicts on a vector when that vector is multiplied by the matrix.

A number associated with a n-dimensional, square matrix which represents the ratio by which the matrix changes n-dimensional volumes in its vector space. For a two-dimensional matrix then, the determinant is the ratio of the change in area produced by the matrix on a square.

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.