A method of computing the determinant of a square matrix due to Charles Dodgson (1866) (who is more famous under his pseudonym
Lewis Carroll). The method is useful for hand calculations because, for an integer
matrix, all entries in submatrices computed along the way must also be integers.
The method is also implemented efficiently in a parallel computation. Condensation
is also known as the method of contractants (Macmillan 1955, Lotkin 1959).

Given an
matrix, condensation successively computes an matrix, an matrix, etc., until arriving at a matrix whose only entry ends up being the determinant
of the original matrix. To compute the matrix (), take the
connected subdeterminants of the matrix and divide them by the central entries of the matrix, with no divisions performed for . The matrices arrived at in this manner are the matrices
of determinants of the
connected submatrices of the original matrices.

For example, the first condensation of the matrix

(1)

yields the matrix

(2)

and the second condensation yields

(3)

which is the determinant of the original matrix. Collecting terms gives

(4)

of which the nonzero terms correspond to the permutation matrices. In the
case, 24 nonzero terms are obtained together with 18 vanishing ones. These 42 terms
correspond to the alternating sign matrices
for which any s
in a row or column must have a "outside" it (i.e., all s are "bordered" by s).

Bareiss, E. H. "Sylvester's Identity and Multistep Integer-Preserving Gaussian Elimination." Math. Comput.22, 565-578,
1968.Bressoud, D. and Propp, J. "How the Alternating Sign Matrix
Conjecture was Solved." Not. Amer. Math. Soc.46, 637-646.Dodgson,
C. L. "Condensation of Determinants, Being a New and Brief Method for Computing
their Arithmetic Values." Proc. Roy. Soc. Ser. A15, 150-155,
1866.Lotkin, M. "Note on the Method of Contractants." Amer.
Math. Soc.55, 476-479, 1959.Macmillan, R. H. A New
Method for the Numerical Evaluation of Determinants." J. Roy. Aeronaut. Soc.59,
772, 1955.Robbins, D. P. and Rumsey, H. Jr. "Determinants
and Alternating Sign Matrices." Adv. Math.62, 169-184, 1986.