A technically defined extension of the ordinary determinant to "higher dimensional" hypermatrices. Cayley originally coined the term, but subsequently used it to refer to an algebraic invariant of a multilinear form. The hyperdeterminant of the 2×2×2 hypermatrix A = a_(i j k) (for i, j, k = 0, 1) is given by det(A) = (a_0^2 a_111^2 + a_1^2 a_110^2 + a_10^2 a_101^2 + a_11^2 a_100^2) - 2(a_0 a_1 a_110 a_111 + a_0 a_10 a_101 a_111 + a_0 a_11 a_100 a_111 + a_1 a_10 a_101 a_110 + a_1 a_11 a_110 a_100 + a_10 a_11 a_101 a_100) + 4(a_0 a_11 a_101 a_110 + a_1 a_10 a_100 a_111). The above hyperdeterminant vanishes iff the following system of equations in six unknowns has a nontrivial solution,

determinant | hypermatrix