This question was motivated by <A HREF="https://mathoverflow.net/q/406753/11260">a recent MO post.</A> 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,$^\ast$ and some experimentation$^{\ast\ast}$ for larger $N$ suggests it is true for all $N$.)

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$^\ast$ For $N=2$ one has $M = \begin{pmatrix}a & b \\ b & c
\end{pmatrix}$, $M^{-1} = \frac{1}{a c - b^2} \begin{pmatrix}c & -b \\ -b & a
\end{pmatrix}$, we know $a,c$ and we know $\beta=b/(ac−b^2)$. There are two solutions for the unknown $b$, $b_\pm=(\pm\sqrt{4ac\beta^2+1}−1)/2\beta$, only $b_+$ gives a positive definite $M$.

$^{\ast\ast}$ <A HREF="https://ilorentz.org/beenakker/MO/pdtest.nb">Mathematica test</A> for $N=3$, when there are 5 solutions for the three unknown matrix elements, but only one of these gives a positive definite $M$.
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