I have a big system of equations with the form: $$\left( \begin{array}[cc]\\ A & B \\ C & D \end{array} \right) \left( \begin{array}[c]\\ X\\Y \end{array} \right) = \left( \begin{array}[c]\\ T_1\\T_2 \end{array} \right),$$ where $A,B,C$ and $D$ are square matrices and $X,Y,T_1,T_2$ are vectors.
Is there a Cramer's-like rule for solving that kind of system?
I'm interested in any theoretical explanation.