  Consider the following parameterized counting variation of the classic subset
sum problem, which arises notably in the context of higher homotopy groups of
topological spaces: Let $\mathbf{v} \in \mathbb{Q}^d$ be a rational vector,
$(T_{1}, T_{2} \ldots T_{m})$ a list of $d \times d$ rational matrices, $S \in
\mathbb{Q}^{h \times d}$ a rational matrix not necessarily square and $k$ a
parameter. The goal is to compute the number of ways one can choose $k$
matrices $T_{i_1}, T_{i_2}, \ldots, T_{i_k}$ from the list such that $ST_{i_k}
\cdots T_{i_1}\mathbf{v} = \mathbf{0} \in \mathbb{Q}^h$.
  In this paper, we show that this problem is $\# W[2]$-hard for parameter $k$.
As a consequence, computing the $k$-th homotopy group of a $d$-dimensional
1-connected topological space for $d > 3$ is $\# W[2]$-hard for parameter $k$.
We also discuss a decision version of the problem and its several modifications
for which we show $W[1]/W[2]$-hardness. This is in contrast to the
parameterized $k$-sum problem, which is only $W[1]$-hard (Abboud-Lewi-Williams,
ESA'14). In addition, we show that the decision version of the problem without
parameter is an undecidable problem, and we give a fixed-parameter tractable
algorithm for matrices of bounded size over finite fields, parameterized the
matrix dimensions and the order of the field.
