Unique solution for circuits in Linear Algebra A standard application of Linear Algebra is circuits and Kirchhoff's Laws. Does anyone know of a proof of uniqueness of a solution of a system given by these laws? There are many, many examples, but little theory regarding why there is always a unique solution.
For reference (Wikipedia)


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*At any node (junction) in an electrical circuit, the sum of currents flowing into that node is equal to the sum of currents flowing out of that node

*The algebraic sum of the products of the resistances of the conductors and the currents in them in a closed loop is equal to the total emf available in that loop.
My thoughts are as follows:
Initially these laws setup two systems $Ax = 0$ and $Bx= b$, respectively. If there are $n$ nodes and m currents, then $A$ is a $n \times m$ matrix. If there are l loops in the circuit, then $B$ is a $l \times m$ matrix. I tried to work with the augmented matrix 
$$  \left[\begin{matrix}  A \\B\end{matrix} \right|\left.\begin{matrix}0\\b\end{matrix}\right]$$
But I see no reason why this always has a unique solution. 

There is a statement that works for your purposes (proposition 9.4) presented in Markov Chains and Mixing Times (Levin, Peres, Wilmer) that you can follow by reading pages 115-118.
Perhaps there are other, better references for you purpose, but this is the first that comes to mind.
