A counterexample: $$A=\left( \begin{array}{ccc} 3 & -3 & 4 \\ -3 & 3 & -4 \\ 4 & -4 & 4 \\ \end{array} \right),\quad B=\left( \begin{array}{ccc} 3 & -3 & 0 \\ -3 & 3 & -4 \\ 0 & -4 & 4 \\ \end{array} \right).$$$$A=\left( \begin{array}{ccc} 4 & 4 & -4 \\ 4 & 6 & -5 \\ -4 & -5 & 6 \\ \end{array} \right),\quad B=\left( \begin{array}{ccc} 4 & 4 & 0 \\ 4 & 6 & -5 \\ 0 & -5 & 6 \\ \end{array} \right).$$ Then $A$ is positive definite but $B$ is not.
