 Solution

False. Every entry of $A=\left(\begin{array}{cc}1& 2\\ 2& 1\end{array}\right)$ is strictly positive, yet $A(1,-1{)}^T=-(1,-1{)}^T$. Its eigenvalues are $3$ and $-1$, so one has negative real part. Positivity of entries does not imply the positive-definite matrix property.