This question was motivated by a recent MO post. You know $n$ elements of the $N\times N$ matrix $M$ and you do not know $n$ elements of the inverse $M^{-1}$ (but you know the other $N^2-n$ elements of $M^{-1}$). 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. (For $N=2$ it is true, and some experimentation for larger $N$ suggests it is true for all $N$.)