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)
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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)
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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)
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in which case is called a cofactor.
The equation for the determinant can also be formally written as
(4)
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where ranges over all permutations of and is the inversion number of (Bressoud and Propp 1999).