Thread: General solution for diagonal zero matrices?

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.

I know if the A matrix is non-singular then Cramer's rule would give a formula for the solution. What I am wondering is whether this nice looking matrix is known to have some nice properties, such as conditions that guarantee for full rank or not...

No, it is easy to make up examples that have determinant 0 and others that do not. I would recommend ignoring the "0"s and just solving as you would any matrix.