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.
- 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:
Using Laplace's formula for the determinant