  We study the first passage time $\tau_u = \inf \{ n \geq 1: |V_n| > u \}$ for
the multivariate perpetuity sequence $V_n = Q_1 + M_1 Q_2 + \cdots + (M_1
\ldots M_{n-1}) Q_n$, where $(M_n, Q_n)$ is a sequence of independent and
identically distributed random variables with $M_1$ a $d \times d$ ($d \geq 1$)
random matrix with nonnegative entries, and $Q_1$ a nonnegative random vector
in $\mathbb R^d$. The exact asymptotic for the probability $\mathbb P (\tau_u
<\infty)$ as $u \to \infty$ has been found by Kesten (Acta Math. 1973). In this
paper we prove a conditioned weak law of large numbers for $\tau_u$:
conditioned on the event $\{ \tau_u < \infty \}$, $\frac{\tau_u}{\log u}$
converges in probability to a certain constant $\rho > 0$ as $u \to \infty$. A
conditioned central limit theorem for $\tau_u$ is also obtained. We further
establish precise large deviation asymptotics for the lower probability
$\mathbb P (\tau_u \leq (\beta - l) \log u)$ as $u \to \infty$, where $\beta
\in (0, \rho)$ and $l \geq 0$ is a vanishing perturbation satisfying $l \to 0$
as $u \to \infty$. Even in the case $d = 1$, this improves the previous large
deviation result by Buraczewski et al. (Ann. Probab. 2016). As consequences, we
deduce exact asymptotics for the pointwise probability $\mathbb P (\tau_u =
[(\beta - l) \log u] )$ and the local probability $\mathbb P (\tau_u - (\beta -
l) \log u \in (a, a + m ] )$, where $a<0$ and $m \in \mathbb Z_+$. We also
establish analogous results for the first passage time $\tau_u^y = \inf \{ n
\geq 1: \langle y, V_n \rangle > u \}$, where $y$ is a nonnegative vector in
$\mathbb R^d$ with $|y| = 1$.
