  We provide a complete study of existence and uniqueness (uniqueness up to
multiples in the case $\mathbf{d} = \mathbf{0}$) of non-negative and
non-trivial solutions $\mathbf{x}$ for the linear system $(\mathbf{I}-
\mathbf{A})\mathbf{x} = \mathbf{d}$ with $\mathbf{A} \geq \mathbf{0},
\mathbf{d} \geq 0$ (which, in particular, applies to Leontief's Input Output
Model). This study is done in terms of the block triangular form of the matrix
$\mathbf{A}$ and is related to a directed graph associated to $\mathbf{A}$. In
particular, this study of existence and uniqueness up to multiples of solutions
provides a framework that allows us to rigorously perform a sensitivity
analysis of the normalized solutions of the linear system $(\mathbf{I} -
\mathbf{A})\mathbf{x} = \mathbf{d}$.
