Umm, what did you try? LinearSolve works like the documentation states. For example:
A = {{1, 2}, {3, 0}};
B = {{11}, {3}};
LinearSolve[A, B]
(* Out[186]= {{1}, {5}} *)
This agrees with multiplying on the left by the inverse matrix.
Inverse[A].B
(* Out[187]= {{1}, {5}} *)
