Something like this?
n = 4;
s = Delete[Table[i, {i, -10, 10}], 11];
{a, b, c, f} = {RandomChoice[s, n], RandomChoice[s, n - 1],
RandomChoice[s, n - 1], RandomChoice[s, n]}
A = SparseArray[{Band[{1, 1}] -> a, Band[{2, 1}] -> b,
Band[{1, 2}] -> c}, {n, n}];
MatrixForm[A]
(* Calculate coefficients \[Alpha] and \[Beta] *)
alpha = Table[Symbol["\[Alpha]" <> ToString[i]], {i, n}];
beta = Table[Symbol["\[Beta]" <> ToString[i]], {i, 2, n}];
alpha[[1]] = a[[1]];
Do[beta[[i - 1]] = b[[i - 1]]/alpha[[i - 1]];
alpha[[i]] = a[[i]] - beta[[i - 1]] c[[i - 1]], {i, 2, n}]
L = SparseArray[{Band[{1, 1}] -> 1, Band[{2, 1}] -> beta}, {n, n}];
U = SparseArray[{Band[{1, 1}] -> alpha, Band[{1, 2}] -> c}, {n, n}];
(* Check to see that L.U \[Equal] A *)
L.U // MatrixForm
LinearSolve[U, LinearSolve[L, f]] // MatrixForm
LinearSolve[A, f] // MatrixForm
