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

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 169-170,
1985.Bressoud, D. and Propp, J. "How the Alternating Sign Matrix
Conjecture was Solved." Not. Amer. Math. Soc.46, 637-646, 1996.Muir,
T. "Minors and Expansions." Ch. 4 in A
Treatise on the Theory of Determinants. New York: Dover, pp. 53-137,
1960.