  The matching complex of a graph $G$ is a simplicial complex whose simplices
are matchings in $G$. In the last few years the matching complexes of grid
graphs have gained much attention among the topological combinatorists. In
2017, Braun and Hough obtained homological results related to the matching
complexes of $2 \times n$ grid graphs. Further in 2019, Matsushita showed that
the matching complexes of $2 \times n$ grid graphs are homotopy equivalent to a
wedge of spheres. In this article we prove that the matching complexes of
$3\times n$ grid graphs are homotopy equivalent to a wedge of spheres. We also
give the comprehensive list of the dimensions of spheres appearing in the
wedge.
