Consider the 1D Poisson equation $$\nabla^2 u = f.$$ Using finite difference method on cell corner data and a uniform grid with ghost points, I think we can write the system of equations with Neumann BCs as: $$Au = f-Au_{BC},$$ where $$ \begin{aligned} A &= \frac{1}{\Delta x^2} \left[\begin{array}{ccccccccc} -2 & 2 & & & & & \\ 1 & -2 & 1 & & & & \\ & & \ddots & \ddots & \ddots & & \\ & & & & 1 & -2 & 1 \\ & & & & & 2 & -2 \\ \end{array} \right] \\ Ax_{BC} &= \frac{1}{\Delta x^2} \left[\begin{array}{ccccccccc} 2 \hat{n} \theta \Delta x \\ 0 \\ \\ \vdots \\ \\ 0 \\ 2 \hat{n} \theta \Delta x \\ \end{array} \right] \end{aligned} $$
And $\hat{n},\theta$ is the outward facing normal and the prescribed slope of the derivative at the boundaries.
Notably, the system is singular, which can be addressed by removing the mean of the RHS.
Question
I know that $A$ must be symmetric, positive definite. I've done some tests without multiplying the first and last rows by 0.5 and they seem to work fine, so my question is:
Must the first and last rows of the left and right hand side be multiplied by 0.5? In other words, are there cases where it will not work without multiplication of 0.5?
Notes
I imagine that both sides of the equation would be balanced the same with and without this multiplication.
As a final note, I know that the code / algorithm may be more clear with the multiplication, since the equation explicitly satisfies symmetry, but I'm more interested in whether the multiplication is necessary or not.
Any help is greatly appreciated.