  This article studies the scheme structure of the jet schemes of determinantal
varieties. We show that in general, these jet schemes are not irreducible. In
the case of the determinantal variety $X$ of $r \times s$ matrices of rank at
most one, we give a formula for the dimension of each of the components of its
jet schemes. As an application, we compute the log canonical threshold of the
pair $(\mathbb{A}^{rs},X)$.
