Let's assume a square matrix M$\in \mathcal{M}(\mathbb{R_0^+})^{n\times n}$, meaning its generic entry $m_{i,j} >0$ for any $i,j=1,\dots,n$, and moreover M is invertible. Is it possible to demonstrate that if $\sum_{i}\sum_{j}m_{i,j}=1$, then $\sum_{j=1}^nm_{i,j}=\sum_{j=1}^nm_{j,i}$ for any chosen $i \in \{1,\dots,n\}$? Which theoretical result may I use? I think that it would be possible to demonstrate it by induction, but I'm more interested in linking it with existing theory results which at the moment I cannot recall. Thanks in advance.
Leonardo Angeli