Determinant Expansion by Minors

Also known as "Laplacian" determinant expansion by minors, expansion by minors is a technique for computing the determinant of a given square matrix M. Although efficient for small matrices, techniques such as Gaussian elimination are much more efficient when the matrix size becomes large.

Let |A| denote the determinant of an n×n matrix A, then for any value i=1, ..., n,


where M_(ij) is a so-called minor of A, obtained by taking the determinant of A with row i and column j "crossed out."


For example, for a 3×3 matrix, the above formula gives

 |a_(11) a_(12) a_(13); a_(21) a_(22) a_(23); a_(31) a_(32) a_(33)|=a_(11)|a_(22) a_(23); a_(32) a_(33)|-a_(12)|a_(21) a_(23); a_(31) a_(33)|+a_(13)|a_(21) a_(22); a_(31) a_(32)|.

The procedure can then be iteratively applied to calculate the minors in terms of subminors, etc. The factor (-1)^(i+j) is sometimes absorbed into the minor as


in which case C_(ij) is called a cofactor.

The equation for the determinant can also be formally written as


where pi ranges over all permutations of {1,2,...,n} and I(pi) is the inversion number of pi (Bressoud and Propp 1999).

See also

Cofactor, Condensation, Determinant, Gaussian Elimination

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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.

Referenced on Wolfram|Alpha

Determinant Expansion by Minors

Cite this as:

Weisstein, Eric W. "Determinant Expansion by Minors." From MathWorld--A Wolfram Web Resource.

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