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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.