Determinant of the sum of two matrices with positive determinant

Question

Let $A = \left( \matrix{a & b \\ c & d}\right)$ and $A^\prime = \left( \matrix{a^\prime & b^\prime \\ c^\prime & d^\prime}\right)$ be $2\times 2$ matrices with positive integer entries such that $\det A > 0$ and $\det A^\prime > 0$.

If $\det \left( \matrix{a + c & a^\prime + c^\prime \\ b + d & b^\prime + d^\prime} \right) = 0$, then is it necessary that $\det (A + A^\prime) \ge 0$?

Answer 1

We can rewrite the conditions e.g. with

$\displaystyle b=\frac{b'+d'}{a'+c'}a-\frac{\det A}{a+c} \enspace$ and $\enspace\displaystyle d=\frac{b'+d'}{a'+c'}c+\frac{\det A}{a+c} \enspace$ .

With $\det A>0$ and $\det A'>0$ and positive real entries it follows

$\displaystyle \det(A+A')=(1+\frac{a'+c'}{a+c})\det A+(1+\frac{a+c}{a'+c'})\det A'>0\enspace$ .