How to prove $rank(A^+)$ is no more than rank(A)?

Question

Look here, my friend. How to prove the following equation? Or give a counter-example.

$$\text{rank}(A^+)\leq \text{rank}(A)$$ where $A$ has full rank, and $\text{rank}(A^+)$ represents the positive components of matrix $A$, e.g. $$\left[\begin{matrix}2 & -1.5 & 0\\-2.3 & 2 & 4.1\end{matrix}\right]^+=\left[\begin{matrix}2 & 0 & 0\\0 & 2 & 4.1\end{matrix}\right].$$ Thank you in advance.

Answer 1

Counter example: $A = \begin{pmatrix} -1 & 1 \\ 1 & -1 \end{pmatrix}$.