During my research I came across the following problem. Intuitively this should be an easy one. However, the simplest version of it looks like this:
Let $C \geq \frac{1}{2}$ be some fixed universal constant and let $\varepsilon > 0$ be given such that
$$C \varepsilon > ac - bd > 0, \quad \text{where} \quad 0 < a,b,c,d \leq \frac{1}{2}.$$
Thus we are concerned with a non-singular $2 \times 2$ matrix with small positive entries and a small determinant. Now I want to perturb the numbers $a,b,c,d$ as small as possible such that we have zero determinant.
More precisely: I want
$$(a + x)(c + y) - (b + z)(d + w) = 0, \quad \text{where} \quad |x|+|y|+|z|+|w| \leq \varepsilon.$$ Additionally I want the new matrix to be still positive: $(a + x),(c + y),(b + z),(d + w) > 0 $.
Moreover I should emphasize that I'm mostly concerned with showing the existance of such $x,y,z,w$ for a given $\varepsilon$ than really computing them or finding explicit minimal ones.
Any help would be highly appreciated!