# linear algebra – ideas how to efficiently tackle a set of equations

I m working on a project, where a set of equations has come up. I ve rewritten these equations into matrix form, which has given me the following matrix equation (i m only writing the left hand side because it s the only problematic part, the right hand side is just a bunch of matrices that all have known quantities and are not that tricky to deal with). I would like to calculate the a terms out of this equation (all the b and c terms are known), however as you can probably see, I can t just combine all the desired terms into one matrix and then solve. This problem could be tackled by just going back to the initial equations and just plugging one equation into another until i get the expression for each a term, however this is inefficient and I ll only consider it as a last resort. So I want to know if anyone could give me some suggestions or advice how to tackle this problem in the most efficient manner possible. Thanks in advance.

$$begin{bmatrix} (1-{b}_{++}c_+)& 0\ 0& (1-b_{–}c_-)\ end{bmatrix}begin{bmatrix} a_{++}& a_{+-}\ a_{-+}& a_{–}\ end{bmatrix}-begin{bmatrix} b_{++}c_+& b_{+-}c_-\ b_{-+}c_+& b_{–}c_-\ end{bmatrix}begin{bmatrix} a_{++}& 0\ 0& a_{–}\ end{bmatrix}=something$$