The system of equations is the following:

$$ \frac{1}{\Gamma} = A\Gamma $$

where Γ is a vector of size n and A is a matrix of size n * n. With n > 100.
(I hope I am writing it correctly. So on the lhs we have the element-wise inverse values of Γ)

So there is a paper showing, that successive substitution is faster than newton-raphson if solved as mentioned above.

I was wondering if through some kind of linear algebra change the system of equations becomes easier / faster to solve?

I don't know it it helps, but here is how the matrix A is calculated: $$ A = B \circ D $$ Where B is symmetric, dense with positive and negative entries and D is dense with only positive entries and all rows are the same and their sum is 1.