This question was motivated by a recent MO post. You know $n$ elements of the $N\times N$ matrix $M$ and you know $N^2-n$ elements of the inverse $M^{-1}$. Equating $(M^{-1})^{-1}=M$ gives $n$ nonlinear equations in $n$ unknowns, which in general will have multiple solutions. Under which additional condition can one reconstruct the matrix $M$ uniquely? Does it matter where in the matrix are the $n$ elements located?

Conjecture: For $n=N$ elements on the diagonal the reconstruction is unique if $M$ is positive definite.