The determinant of a 2 × 2 matrix is denoted either by "det" or by vertical bars around the matrix, and is defined as
For … See more

If the matrix entries are real numbers, the matrix A can be used to represent two linear maps: one that maps the standard basis vectors to the rows of A, and one that maps them to the columns of A. In either case, the images … See more

Let A be a square matrix with n rows and n columns, so that it can be written as
The entries etc. are, for many purposes, real or complex numbers. As discussed below, the determinant is also defined for matrices w… See more

The determinant can be characterized by the following three key properties. To state these, it is convenient to regard an -matrix A as being composed of its columns, so denoted as
where the … See more

The determinant is closely related to two other central concepts in linear algebra, the eigenvalues and the characteristic polynomial of a matrix. Let be an -matrix with complex entries. Then, by the Fundamental Theorem o… See more

Historically, determinants were used long before matrices: A determinant was originally defined as a property of a system of linear equations. The determinant "determines" whether the system has a unique solu… See more

Determinants can be used to describe the solutions of a linear system of equations, written in matrix form as . This equation has a unique solution if and only if is nonzero. In this case, the solution is given by Cramer's rule… See more