A square matrix with real entries is called balanced if the sum of the entries in each row is zero. Show that every cofactor of a balanced matrix M is the same.
I'd like to prove this statement. First, I'd like to denote that this definition of balanced matrix might be narrowly defined from the conventional usage of the balanced matrix.
I want to prove it through graph theory, where this balanced matrix refers to the incidence matrix of directed graph where each vertex has the same number of inflow and outflow edges.
Now I need to know what cofactor refer to in incidence matrix. What would the cofactor mean in graphical notation of matrix? And How this mean would lead me to the proof where cofactors are all same?
Any hint or guidance would be appreicated.