A counterexample: 
$$A=\left(
\begin{array}{ccc}
 3 & -3 & 4 \\
 -3 & 3 & -4 \\
 4 & -4 & 4 \\
\end{array}
\right),\quad
B=\left(
\begin{array}{ccc}
 3 & -3 & 0 \\
 -3 & 3 & -4 \\
 0 & -4 & 4 \\
\end{array}
\right).$$
Then $A$ is positive definite but $B$ is not.