For a given matrix, return the determinant and the permanent of the matrix.

The determinant is given by

 



det
(
A
)
=

∑

σ


sgn
⁡
(
σ
)

∏

i
=
1


n



M

i
,

σ

i






{\displaystyle \det(A)=\sum _{\sigma }\operatorname {sgn}(\sigma )\prod _{i=1}^{n}M_{i,\sigma _{i}}}


while the permanent is given by

 



perm
⁡
(
A
)
=

∑

σ



∏

i
=
1


n



M

i
,

σ

i






{\displaystyle \operatorname {perm} (A)=\sum _{\sigma }\prod _{i=1}^{n}M_{i,\sigma _{i}}}


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.



Related task

 Permutations by swapping


const determinant = arr =>
    arr.length === 1 ? (
        arr[0][0]
    ) : arr[0].reduce(
        (sum, v, i) => sum + v * (-1) ** i * determinant(
            arr.slice(1)
            .map(x => x.filter((_, j) => i !== j))
        ), 0
    );
 
const permanent = arr =>
    arr.length === 1 ? (
        arr[0][0]
    ) : arr[0].reduce(
        (sum, v, i) => sum + v * permanent(
            arr.slice(1)
            .map(x => x.filter((_, j) => i !== j))
        ), 0
    );
 
const M = [
    [0, 1, 2, 3, 4],
    [5, 6, 7, 8, 9],
    [10, 11, 12, 13, 14],
    [15, 16, 17, 18, 19],
    [20, 21, 22, 23, 24]
];
console.log(determinant(M));
console.log(permanent(M));