Prove that sum of a positive matrix and a positive definite matrix is invertible

Question

Let $A$ be a positive matrix, and $B$ be a positive definite matrix. Prove that $A+B$ is invertible.

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I tried to use the prove of positive definite implies invertible, but there is nothing I can use from the property of being a positive matrix.

Answer 1

The statement is incorrect. Take $$A=\begin{pmatrix}1&2\\2&1\end{pmatrix}, \ \ B=\begin{pmatrix}1&0\\0&1\end{pmatrix}.$$ Then $$A+B=\begin{pmatrix}2&2\\2&2\end{pmatrix}$$ is singular.