In linear algebra, the

**permanent**of an

*n*-by-

*n*matrix

*A*=(

*a*

_{i,j}) is defined as

_{n}, i.e. over all permutations of the number 1,2,...,

*n*.

For example,

*A*differs from that of the determinant of

*A*in that the signatures of the permutations are not taken into account. If one views the permanent as a map that takes

*n*vectors as arguments, then it is a multilinear map and it is symmetric (meaning that any order of the vectors results in the same permanent). A formula similar to Laplace's for the development of a determinant along a row or column is also valid for the permanent; all signs have to be ignored for the permanent.

Unlike the determinant, the permanent has no easy geometrical interpretation; it is mainly used in combinatorics. The permanent is not multiplicative. It is also not possible to use Gaussian elimination to compute the permanent; no fast algorithms for its computation are known.