The determinant of the matrix
can be written
where
is any permutation of
and
if the permutation is even and -1 if the permutation is odd, and matching raised and lower indices indicate summation.
In the last line 123,231,312 are even, and are added.
132, 213, 321 are odd and are subtrated.
This result can be generalised.
For a square matrix
of order
