Solving matrix in mathematica into a linear system

Question

A = SparseArray[{Band[{1, 2}] -> 1, Band[{2, 1}] -> -1}, {n, 
     n}]/(2*\[CapitalDelta]x);

B = SparseArray[{Band[{1, 1}] -> -2, Band[{1, 2}] -> 1, 
     Band[{2, 1}] -> 1}, {n, n}]/(\[CapitalDelta]x^2);

R = SparseArray[{Band[{1, 1}] -> 1}, {n, n}];

pde = -D[V[S, t], t] == ((r - 0.5 \[Sigma]^2).A + (0.5 \[Sigma]^2).B - r.R)V

Having constructed my matrices A,B and R, how do I solve in mathematica to get a single matrix? I need to replace it in the above equation. I've tried the above codes but failed. I need every element in each matrix be multiplied to the respective scalar. 

Answer 1

((r - 0.5 \[Sigma]^2)*A + (0.5 \[Sigma]^2)*B - r*R).Table[
   V[s, t][[i]], {i, 1, n}] 

So Mathematica sees the different dimensions of V.

It behaves like this:

IdentityMatrix[3].y
(* {{1,0,0},{0,1,0},{0,0,1}}.y *)

IdentityMatrix[3].{y[1], y[2], y[3]}
(* {y[1],y[2],y[3]} *)