The convergence of power matrices

For the class of $d\times d$ matrices $B=[b_{i,j}]$ with complex nonzero entries satisfying ${\sum }_{i=1}^d\mathrm{\mid }{b}_{i,j}\mathrm{\mid }=1$, we provide the conditions for the convergence of power matrices $B^n$ to a nonzero limit matrix. In particular, for $d\ge 3$ we prove that the sequence of matrices $B^n$ converges to a nonzero limit matrix if and only if all the entries are positive and real.