You can build the system matrix as follows

A = RandomReal[{-1, 1}, {2, 2}];
A = A\[Transpose].A;
n = 10;
AA = Plus[
  KroneckerProduct[SparseArray[{Band[{2, 1}] -> -1, Band[{1, 2}] -> 1}, {n, n}], A],
  KroneckerProduct[IdentityMatrix[n, SparseArray], N@IdentityMatrix[2]]
  ]

The right-hand side can be assembled as follows:

d = RandomReal[{-1, 1}, {n, 2}];
S1 = RandomReal[{-1, 1}, {2}];
Sn = RandomReal[{-1, 1}, {2}];
b = Flatten[d];
b[[;; 2]] += A.S1;
b[[-2 ;;]] += A.Sn;

Solve as usual (this is a usual linear system). I'd suggest a solver for banded matrices:

LinearSolve[AA, b, Method -> "Banded"]

You can build the system matrix as follows

A = RandomReal[{-1, 1}, {2, 2}];
A = A\[Transpose].A;
n = 10;
AA = Plus[
  KroneckerProduct[SparseArray[{Band[{2, 1}] -> -1, Band[{1, 2}] -> 1}, {n, n}], A],
  KroneckerProduct[IdentityMatrix[n, SparseArray], N@IdentityMatrix[2]]
  ]

The right-hand side can be assembled as follows:

d = RandomReal[{-1, 1}, {n, 2}];
S1 = RandomReal[{-1, 1}, {2}];
Sn = RandomReal[{-1, 1}, {2}];
b = Flatten[d];
b[[;; 2]] += A.S1;
b[[-2 ;;]] -= A.Sn;

Solve as usual (this is a usual linear system). I'd suggest a solver specialized for banded matrices:

x = LinearSolve[AA, b, Method -> "Banded"];

In order to convert back to a list of pairs, you can use

Partition[x,2]