In fact, this determinant condition is satisfied identically because the antisymmetry in the second pair of indices on χ χ shows that A ^ ^ ∝ ∝ k is already a trivial solution. Therefore, any non-trivial solutions must reside in the 3-dimensional subspace orthogonal to k , so the tensor k ♯ ♯ ⋅ ⋅ χ χ ⋅ ⋅ k is effectively only 3-dimensional. Thus, the determinant condition is insufficient to provide any information. However, the classical adjugate of a matrix M is related to its determinant by M . a d j ( M ) = det ( M ) I . Since in this case det ( M ) = 0 but M is arbitrary, one obtains the secondary condition