  In this paper, we show that a locally constant cocycle $\mathcal{A}$ is
$k$-quasi multiplicative under the irreducibility assumption. More precisely,
we show that if $\mathcal{A}^t$ and $\mathcal{A}^{\wedge m}$ are irreducible
for every $t \mid d$ and $1\leq m \leq d-1$, then $\mathcal{A}$ is
$k$-uniformly spannable for some $k\in \mathbb{N}$, which implies that
$\mathcal{A}$ is $k$-quasi multiplicative. We apply our results to show that
the unique subadditive equilibrium Gibbs state is $\psi$-mixing and calculate
the Hausdorff dimension of cylindrical shrinking target and recurrence sets.
