We consider the family of matrices $B = \left[ b _ { i , j } \right] _ { 1 \leq i , j \leq n } \in \mathcal { M } _ { n } ( \mathbb { R } )$ satisfying the following three properties (called $M$-matrices): $$\forall i \in \{ 1 , \ldots , n \} , \left\{ \begin{array} { l }
b _ { i , i } > 0 \\
b _ { i , j } \leq 0 \text { for all } j \neq i \\
\sum _ { j = 1 } ^ { n } b _ { i , j } > 0
\end{array} \right.$$ Show that if $B$ is an $M$-matrix, then we have (a) $B$ is invertible (b) If $F = {}^{ t } \left( f _ { 1 } , \ldots , f _ { n } \right)$ has all positive coordinates, then $B ^ { - 1 } F$ also, (c) all coefficients of $B ^ { - 1 }$ are positive.
We consider the family of matrices $B = \left[ b _ { i , j } \right] _ { 1 \leq i , j \leq n } \in \mathcal { M } _ { n } ( \mathbb { R } )$ satisfying the following three properties (called $M$-matrices):
$$\forall i \in \{ 1 , \ldots , n \} , \left\{ \begin{array} { l }
b _ { i , i } > 0 \\
b _ { i , j } \leq 0 \text { for all } j \neq i \\
\sum _ { j = 1 } ^ { n } b _ { i , j } > 0
\end{array} \right.$$
Show that if $B$ is an $M$-matrix, then we have
(a) $B$ is invertible
(b) If $F = {}^{ t } \left( f _ { 1 } , \ldots , f _ { n } \right)$ has all positive coordinates, then $B ^ { - 1 } F$ also,
(c) all coefficients of $B ^ { - 1 }$ are positive.