  Consider a random $n\times m$ matrix $A$ over the finite field of order $q$
where every column has precisely $k$ nonzero elements, and let $M[A]$ be the
matroid represented by $A$. In the case that q=2, Cooper, Frieze and Pegden
(RS\&A 2019) proved that given a fixed binary matroid $N$, if $k\ge k_N$ and
$m/n\ge d_N$ where $k_N$ and $d_N$ are sufficiently large constants depending
on N, then a.a.s. $M[A]$ contains $N$ as a minor. We improve their result by
determining the sharp threshold (of $m/n$) for the appearance of a fixed
matroid $N$ as a minor of $M[A]$, for every $k\ge 3$, and every finite field.
