Clear["Global'*"]
l = {{0, 0, 0, 0}, {-1, 0, 0, 0}, {-1, -1, 0, 0}, {-1, -1, -1, 0}};
d = {{5, 0, 0, 0}, {0, 10, 0, 0}, {0, 0, 5, 0}, {0, 0, 0, 10}};
u = {{0, -1, -1, -1}, {0, 0, -1, -1}, {0, 0, 0, -1}, {0, 0, 0, 0}};
b = {{4}, {12}, {8}, {34}};
B = N[Dot[-Inverse[d], l + u]];
d = Dot[Inverse[d], b];
x[0] = {{0}, {0}, {0}, {0}};
\[Delta][0] = "\[Delta]";
Do[{x[i] = Dot[B, x[i - 1]] + d, \[Delta][i] =
Max[Abs[x[i] - x[i - 1]]], k = i} ;
If [\[Delta] < 10^4, Break[]], {i, 1, \[Infinity]}];
Table[{i, NumberForm[x[i][[1, 1]], {\[Infinity], 5}],
NumberForm [x[i][[2, 1]], {\[Infinity], 5}],
NumberForm[x[i][[3, 1]], {\[Infinity], 5}],
NumberForm[x[i][[4, 1]], {\[Infinity], 5}],
NumberForm[\[Delta][i], {\[Infinity], 5}]}, {i, 0,
k}]; MatrixForm[%]