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).
See also
Cofactor, Condensation, Determinant, Gaussian Elimination
Explore with Wolfram|Alpha
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.
Referenced on Wolfram|Alpha
Determinant Expansion by Minors
Cite this as:
Weisstein, Eric W. "Determinant Expansion by Minors." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DeterminantExpansionbyMinors.html