The complexity of computing, via threshold circuits, the iteratedproduct and powering of fixed-dimension $k\times k$
matriceswith integer or rational entries is studied. We call these twoproblems $\sf IMP_k$
and $\sf MPOW_k$
, respectively, for short. We prove that: (i) For $k\geq2$
, $\sf IMP_k$
does not belong to ${\rm TC}^0$
, unless ${\rm TC}^0={\rm NC}^1$
.newline(ii) For stochastic matrices : $\sf IMP_2$
belongs to ${\rm TC}^0$
while, for $k\geq3$
, $\sf IMP_k$
does not belong to ${\rm TC}^0$
, unless ${\rm TC}^0={\rm NC}^1$
. (iii) For any k, $\sf MPOW_k$
belongs to ${\rm TC}^0$
.