Let $$ A = \begin{bmatrix} -a_{12}-a_{13}-a_{14} & a_{12} & a_{13} & 1\\ a_{21} & -a_{21}-a_{23}-a_{24} & a_{23} & 1\\ a_{31} & a_{32} & -a_{31} - a_{32} - a_{34} & 1\\ a_{41} & a_{42} & a_{43} & 1 \end{bmatrix}, $$ where all $a_{ij}$'s are positive reals. If we explicitly calculate the determinant of $A$ and factor whole expression, then we can easily see that $\det(A) < 0$.
Is it possible to prove that $\det(A) < 0$ (or that $\det(A) \neq 0$ if it is easier) using some matrix manipulations without calculating it directly?