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Is the following statement correct?

Let $A$ be a symmetric 0-1 matrix with non-zero determinant, all diagonal elements equal to 0 and an odd number of rows. The diagonal of the cofactor matrix of $A$ contains a non-zero element.

I verified this statement with a computer for millions of matrices $A$ and it seems to be correct, but I am unable to prove it.

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    $\begingroup$ you might replace cofactor matrix of $A$ by $A^{-1}$, right? $\endgroup$
    – Carlo Beenakker
    Commented May 25 at 19:39
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    $\begingroup$ That is correct. You can replace the cofactor matrix of $A$ by $A^{-1}$. $\endgroup$
    – user528497
    Commented May 25 at 20:41

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