  We prove two dichotomy theorems about sequences of operators into $L_1$ given
by random matrices. In the second theorem we assume that the entries of each
random matrix form a sequence of independent, symmetric random variables. Then
the corresponding sequence of operators either uniformly factor the identity
operators on $\ell_1^k$ $(k\in\mathbb N$) or uniformly approximately factor
through $\mathrm{c}_0$. The first theorem has a slightly weaker conclusion
still related to factorization properties but makes no assumption on the random
matrices. Indeed, it applies to operators defined on an arbitrary sequence of
Banach spaces. These results provide information on the closed ideal structure
of the Banach algebra of all operators on the space
$\big(\bigoplus_{n=1}^\infty\ell_1^n \big)_{\mathrm{c}_0}$.
