Linear system with positive semi-definite matrix

Question

I have linear system $Ax=b$, where $A$ is positive semi-definite, with positive elements and symmetric ($A$ is variance-covariance matrix ). I know that vector $b$ has elements $b_1>0$ and the rest $b_i<0$, for all $i=2,\dots,N$. Prove that first component of solution has to be positive, i.e. $x_1>0$. Does anybody have any idea?

Answer 1

I don't think $x_1$ must be positive.

A counter example might be a positive definite matrix $A = [1 \space -0.2 ; \space -0.2 \space 1]$ with its inverse matrix $A^{-1}$ having $A_{11}, A_{12} > 0$.

- Edit: Sorry. A counter example might be a normalized covariance matrix

$ A= \left( \begin{array}{ccc} 1 & 0.6292 & 0.6747 & 0.7208 \\ 0.6292 & 1 & 0.3914 & 0.0315 \\ 0.6747 & 0.3914 & 1 & 0.6387 \\ 0.7208 & 0.0315 & 0.6387 & 1 \end{array} \right) $.