Cauchy's determinant theorem states that any row 
and column 
of a determinant being selected,
if the element common to them be multiplied by its cofactor
in the determinant, and every product of another
element of the row by another element of the columns be multiplied by its cofactor,
the sum of the results is equal to the given determinant.
Symbolically,
 |  |  |
(1)
|
 |  |  |
(2)
|
where 
,
2, ..., 
;
;
;
and the sign before 
is determined by the formula 
, with 
the total number of permutation
inversions in the suffix and 
.
