I'm writing an model in which I have to solve a tedious system of equations. It turns out that, after some work, the system can be transformed into a seemingly nice and simple matrix problem. It takes the form

where all the a's and c's are strictly positive, and the objective is to find the b's.

Does any of you know if there is any general formula for the solution of such matrix (that only the diagonal are zeros)? and if such matrix has a special name? (I tried "zero diagonal matrix" and "diagonal zero matrix" on Google without getting anything useful)

Thanks.