  We consider the question of the boundedness of matrix products
$A_{n}B_{n}\cdots A_{1}B_{1}$ with factors from two sets of matrices,
$A_{i}\in\mathscr{A}$ and $B_{i}\in\mathscr{B}$, due to an appropriate choice
of matrices $\{B_{i}\}$. It is assumed that for any sequence of matrices
$\{A_{i}\}$ there is a sequence of matrices $\{B_{i}\}$ for which the sequence
of matrix products $\{A_{n}B_{n}\cdots A_{1}B_{1}\}_{n=1}^{\infty}$ is norm
bounded. Some situations are described in which in this case the norms of
matrix products $A_{n}B_{n}\cdots A_{1}B_{1}$ are uniformly bounded, that is,
$\|A_{n}B_{n}\cdots A_{1}B_{1}\|\le C$ for all natural numbers $n$, where $C>0$
is some constant independent of the sequence $\{A_{i}\}$ and the corresponding
sequence $\{B_{i}\}$. In the general case, the question of the validity of the
corresponding statement remains open.
