An estimate for the exponent of some sets of nonnegative matrices

The exponent of a set $\mathcal A$ of non-negative $k\times k$ matrices is a minimal $n$ such that for any sample with replacement $A_1,\dots,A_n\in\mathcal A$ all elements of the matrix $A_1\ldots A_n$ are positive. We obtain upper bounds of the exponent of some sets of matrices with the use of singular values of matrices. We also give an estimate of the exponent of a set of matrices obtained with the use of a generalized Kronecker product of matrices. These results are used for estimating the length of the covering of a group by a given set of generators.