How many $2\times2$ positive integer matrices are there with a constant trace and positive determinant? The trace of a $2\times2$ positive integer matrix is a given constant positive value. How many possible choices are there such that the determinant is greater than 0? Each element of matrix is positive.

Infinitely many.  Choose any  positive integer $m\geq3$, then  the matrices $\pmatrix{ m-1 & 1 \cr 1 & 1\cr}$  all have trace $m$ and positive determinant. 
EDIT: After the comment pointing out the mistake, I am embarrassed. The answer is finite, bounded below by $m-1$. 
