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# 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 . 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).

Cofactor, Condensation, Determinant, Gaussian Elimination

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## References

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.

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Determinant Expansion by Minors

## Cite this as:

Weisstein, Eric W. "Determinant Expansion by Minors." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DeterminantExpansionbyMinors.html