  Let $\Gamma$ be the multiplicative semigroup of all $n\times n$ matrices with
integral entries and positive determinant. Let $1\leq p \leq n-1$ and
$V=\R^n\oplus \cdots \oplus \R^n$ ($p$ copies). We consider the componentwise
action of $\Gamma$ on $V$. Let $\bx\in V$ be such that $\Gamma \bx$ is dense in
$V$. We discuss the effectiveness of the approximation of any target point $\by
\in V$ by the orbit $\{ \gamma \bx \mid \gamma \in \Gamma\}$, in terms of
$\norm \gamma \norm$, and prove in particular that for all $\bx$ in the
complement of a specific null set described in terms of a certain Diophantine
condition, the exponent of approximation is $(n-p)/p$; that is, for any
$\rho<(n-p)/p$, $\norm \gamma \bx - \by \norm < \norm \gamma \norm^{-\rho}$ for
infinitely many $\gamma$.
