Also known as "Laplacian" determinant expansion by minors, expansion by minors is a technique for computing the determinant
of a given square matrix 
. Although efficient for small matrices, techniques such as
Gaussian elimination are much more efficient
when the matrix size becomes large.
Let 
denote the determinant of an 
matrix 
, then for any value 
, ..., 
,
 |
(1)
|
where 
is a so-called minor of 
, obtained by taking the determinant of 
with row 
and column 
"crossed out."
For example, for a 
matrix, the above formula gives
 |
(2)
|
The procedure can then be iteratively applied to calculate the minors in terms of subminors, etc. The factor 
is sometimes absorbed into the minor as
 |
(3)
|
in which case 
is called a cofactor.
The equation for the determinant can also be formally written as
 |
(4)
|
where 
ranges over all permutations of 
and 
is the inversion number
of 
(Bressoud and Propp 1999).
