Suppose we have a real, symmetric matrix $A(x_1,x_2,x_3)$ given by \begin{pmatrix} a_{1,1} & a_{1,2} & x_1 & x_2 \\ a_{2,1} & a_{2,2} & a_{2,3} & x_3 \\ x_1 & a_{3,2} & a_{3,3} & a_{3,4} \\ x_2 & x_3 & a_{3,4} & a_{4,4} \end{pmatrix}
We would like to complete this matrix into a positive definite matrix by choosing appropriate values for $x_1,x_2, x_3$. (Assume that the $a$ values permit this by themselves). Many completions are possible.
However, only one unique completion is possible such that its inverse is of the form \begin{pmatrix} b_{1,1} & b_{1,2} & 0 & 0 \\ b_{2,1} & b_{2,2} & b_{2,3} & 0 \\ 0 & b_{3,2} & b_{3,3} & b_{3,4} \\ 0 & 0 & b_{3,4} & b_{4,4} \end{pmatrix} i.e it has zeroes in the positions corresponding to $x_1, x_2, x_3$. Furthermore, apparently these values of $x_1,x_2,x_3$ end up maximizing the determinant of $A$.
How do we prove this, and what are the corresponding values of $x_1, x_2, x_3$?
The relationship between the completion and maximizing determinant seemed clear since the conditions on the inverse are equivalent to the partial derivatives of the determinant function being zero. But I am unable to establish uniqueness.