n = 
        5;

    sns = Table[Sin[((2 π)/12) j l], {j, 1, n}]

    csn = Table[Cos[((2 π)/12) j l], {j, 1, n}]

    scn = 
        Flatten[Table[{sns[[ii]], csn[[ii]]}, {ii, 1, 
              n}]];

    row[l_] = Flatten[Append[{1, (((-1))^l)}, scn]]

MLHS = 
        Table[row[l], {l, 1, 12}];

    MLHS // MatrixForm

MRHS = Table[row[l + 1], {l, 1, 12}]; 
    MRHS // MatrixForm

ILHSTRHS = 
        Inverse[N[MLHS]]  .  MRHS // Chop;
ILHSTRHS // MatrixForm

rhs = Table[{0}, {12}];  
    rhs[[12]] = {1 - T};
Inverse[N[MLHS]]  .  rhs  // Chop // MatrixForm

1/12 // N