Problem Statement

For a given matrix, return the determinant and the permanent of the matrix. The determinant is given by while the permanent is given by In both cases the sum is over the permutations

σ

{\displaystyle \sigma }

of the permutations of 1, 2, ..., n. (A permutation's sign is 1 if there are an even number of inversions and -1 otherwise; see parity of a permutation.) More efficient algorithms for the determinant are known: LU decomposition, see for example wp:LU decomposition#Computing the determinant. Efficient methods for calculating the permanent are not known.

Let's start with the solution: