Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

If $A$ is a $n \times n$ nonsingular matrix, and $\det \left( {\begin{array}{*{20}{c}} {{a_{ij}}} & {{a_{ik}}} \\ {{a_{lj}}} & {{a_{lk}}} \\ \end{array}} \right) = {c_{ijkl}}$ for all $i,j,k,l \in \{ 1,2, \cdots n\}$, then can we solve all $a_{ij}$ in terms of $c_{ijkl}$?

share|cite|improve this question
Naively: we have a system of homogeneous $n^2$-variate polynomials of the form: $x_i x_j - x_k x_l = c_m.$ And then Groebner basis.. – user2468 Mar 16 '12 at 4:34
Presumably $c_{ijkl} = -c_{ljki} = -c_{ikjl}$. – Robert Israel Mar 16 '12 at 7:34

1 Answer 1

up vote 1 down vote accepted

For large $n$ it's a highly overdetermined system, with ${n \choose 2}^2$ equations in $n^2$ unknowns. For $n=2$, on the other hand, there's only one nontrivial equation, which just gives you the determinant of $A$. And of course you can multiply the whole matrix by $-1$ without changing any of those determinants.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.