  We first give two new proofs of an old result that the reduced Euler
characteristic of a matroid complex is equal to the M\"obius number of the
lattice of cycles of the matroid up to the sign. The purpose has been to find a
model to establish an analogous result for the case of $q$-matroids and we find
a relation between the Euler characteristic of the simplicial chain complex
associated to a $q$-matroid complex and the lattice of $q$-cycles of the
$q$-matroid. We use this formula to find the complete homology over
$\mathbb{Z}$ of this shellable simplicial complex. We give a characterization
of nonzero Euler characteristic for such order complexes. Finally, based on
these results we remark why singular homology of a $q$-matroid equipped with
order topology may not be effective to describe the $q$-cycles unlike the
classical case of matroids.
