The determinant of an odd order skew-symmetric matrix is always

If $\mathrm{\Delta }=\left|\begin{array}{ccc}1+a& 1& 1\\ 1& 1+b& 1\\ 1& 1& 1+c\end{array}\right|$, then consider the following statements

$\mathrm{I}.$ If $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0,$ then $\mathrm{\Delta }=abc$.

$\mathrm{II}$. If $a^{-1}+b^{-1}+c^{-1}=-1,$ then $\mathrm{\Delta }=0$.

Which of the above statement(s) is / are correct?