# Determinant

## Determinant(c, I, J)

Computes the determinant of a square matrix, «c». Indexes «`I`» and «`J`» must be the same length, so that the matrix is square.

The determinant of a matrix is a useful value that is used heavily in linear algebra and matrix applications. The inverse of a matrix exists and is unique if and only if the determinant is non-zero.

Geometrically, the determinant can be viewed as the N-dimensional volume of the N-dimensional parallelepiped formed from the vectors in matrix «c». A parallelepiped in the N-dimensional generalization of a parallelogram (a 2-D parallelepiped is a parallelogram). Each vertex of the parallelepiped is obtained by taking a subset of the slices along «`J`» and adding them together. The empty subset is the origin. There are `2^N` subsets, where `N=Size(J)`.

## Examples

The determinant of the identity matrix is 1.

`Determinant(I=J,I,J) → 1`
`Determinant(c,I,J)` → 15
`Determinant(c,I,J)` → -114 - 68j

`Variable Matrix :=`
M ▶
L ▼ 1 2 3 4 5
1 6 2 6 3 1
2 2 4 3 1 3
3 6 3 9 3 4
4 3 1 3 8 4
5 1 3 4 4 7
`Determinant(Matrix,L,M) → 359`