The iteration can be performed as follows:
A[t_] = {{t, 0}, {Cos[t], 1}};
B[t_] = {{Exp[t], 0}, {0, Exp[-t]}};
T[F_] := Expand[A[t] + Integrate[B[s].F /. t -> s, {s, 0, t}]];
iteration = NestList[T, {{t, 0}, {0, t}}, 7];
Column[Framed /@ MatrixForm /@ iteration[[1 ;; 3]]]

Well, it's not immediately clear what the limits are, as it was over in this answer, but we can examine the plots to see if we appear to have convergence.
funcs[i_, j_] := Table[iteration[[k, i, j]], {k, 1, Length[iteration]}];
joined = {{funcs[1, 1], funcs[1, 2]}, {funcs[2, 1], funcs[2, 2]}};
GraphicsGrid[Map[Plot[#, {t, -1, 1}] &, joined, {2}]]

As Alexei suggests, it's easy enough to solve the corresponding set of differential equations and compare the results. The solutions are not elementary, however, and involve integrals of special functions.