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.
A matrix over a commutative ringR 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: