M = {{1, 0, 0}, {-1, 1, 0}, {1, 
          0, -1}}; 
L = Inverse[M]; 
MatrixForm[L]

F = {{(1/(1 - w)), ((-w)/(((1 - w))^2)), 
          (w/(1 - (w^2)))}, 
{((-w)/(((1 - w))^2)), 
          w (((1 + w))/(((1 - w))^3)), -w (((1 + (w^2)))/(((1 - (w^2)))^2))}, 
{(w/((1 - (w^2)))), -w (((1 + (w^2)))/(((1 - (w^2)))^2)), (w/((1 - (w^2))))}}; 
MatrixForm[F]

FInv = Inverse[F] // Simplify; 
MatrixForm[FInv]

f0  =  {{1}, {0}, {0}};

FInvTimesf0  = FInv  .  f0; 
MatrixForm[FInvTimesf0]

Limit[FInvTimesf0, w -> 0]

LT = Transpose[L]; 
fStar = {{1}, {(1/2)}, {((-1)/2)}};

fTimesOnes  =  fStar  .  {{1, 1, 1}};

LT - fTimesOnes // MatrixForm

bdSolution = 
      RSolve[ {b[
              n + 1] == (1/2) (b[n] - d[n]) + (1/2) 
                z[n], 
d[
              n + 1] == (1/2) (b[n] - d[n]) - (1/2) z[n]}, {b[
            n], d[n]}, n]

b[n] + d[n]  /.  bdSolution  //  
      Simplify

b[n]  /.  bdSolution  //  
      Simplify