Let $A=(a_{ij})_{i,j=1}^n$ be a positive definite matrix with $a_{ij}>0$ for all $i, j$. Define the Hadamard inverse $A^{\circ 1}$ of $A$ as $(a_{ij}^{1})_{i,j=1}^n$. Is it possible to decide whether $\det A^{\circ 1}$ is positive or negative?

$\begingroup$ The body asks for the determinant of the inverse, but the title asks for the determinant of the product of the matrix and the inverse. Please edit for consistency. $\endgroup$ – Gerry Myerson Feb 12 '17 at 12:12

1$\begingroup$ @Gerry Myerson: Not any more. $\endgroup$ – JanChristoph SchlagePuchta Feb 12 '17 at 12:29
I'm not sure what kind of answer you're looking for. There are $4\times4$ examples of positive definite symmetric matrices where the determinant of the Hadamard inverse is negative and examples where it is positive.
For the first type, take a $4\times4$ matrix that with all entries 1, and replace the main diagonal with 100's.
For the second type, start with the above and then replace the "antidiagonal" (the entries (1,4), (2,3), (3,2), (4,1)) with 1/100.
Of course you can still decide if the determinant is positive or negative by computing it.

$\begingroup$ Is your second type positive definite? Doesn't it have a negative determinant? $\endgroup$ – Gerry Myerson Feb 12 '17 at 22:12

$\begingroup$ It still has the 100's on the main diagonal $\endgroup$ – Anthony Quas Feb 13 '17 at 1:16


$\begingroup$ but the second matrix is not positive definite. $\endgroup$ – Isha Garg Feb 13 '17 at 6:28

$\begingroup$ @IshaGarg: If the second matrix were required to be positive definite, it would automatically have a positive determinant and the question would be moot. $\endgroup$ – Anthony Quas Feb 14 '17 at 2:45