It is known that if \(\{Y_n\}\) is a stationary sequence of random matrices and \(|\cdot |\) a Banach algebra norm, then \[ \lim_{t \to \infty} {1\over t} \log |X_{0t}|= \gamma\;\text{ a.s.},\tag{1} \] where \(X_{0t} \equiv Y_1 Y_2 \dots Y_t\) and \(\gamma\) is a constant under hypotheses of the 0-1 law. The present paper is devoted to a generalization of this result to the case when \(X_{0t}\) to be of trace class can be partitioned in the form \(X_{0t} = \left [\begin{smallmatrix} A_t & B_t\\C_t & D_t\end{smallmatrix} \right]\). For these matrices a relation is derived, which is analogous to (1) and involves the trace-class norms of the off-diagonal parts \(B_t\) and \(C_t\). This relation in the case of Steinhaus random variables \(Y_n\) provides convergence of the off-diagonal terms of \(X_{0t}\) to their mean values.