Linear Algebra/Determinant

The determinant is a function which associates to a square matrix an element of the field on which it is defined (commonly the real or complex numbers). The determinant is required to hold these properties: It is linear on the rows of the matrix. If the matrix has two equal rows its determinant is zero. The determinant of the identity matrix is 1. It is possible to prove that , making the definition of the determinant on the rows equal to the one on the columns. Properties The determinant is zero if and only if the rows are linearly dependent. Changing two rows changes the sign of the determinant: The determinant is a multiplicative map in the sense that for all n-by-n matrices and . This is generalized by the Cauchy-Binet formula to products of non-square matrices. It is easy to see that and thus for all -by- matrices and all scalars . A matrix over a commutative ring R is invertible if and only if its determinant is a unit in R. In particular, if A is a matrix over a field such as the real or complex numbers, then A is invertible if and only if det(A) is not zero. In this case we have Expressed differently: the vectors v1,...,vn in Rn form a basis if and only if det(v1,...,vn) is non-zero. A matrix and its transpose have the same determinant: The determinants of a complex matrix and of its conjugate transpose are conjugate: Theorems Uniqueness Existence Using Laplace's formula for the determinant Binet's theorem Applications