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If $G(M)$ denotes the subgraph of a graph $G$ induced by the set of verticesthat are covered by some matching $M$ in $G$, then $M$ is an induced or auniquely restricted matching if $G(M)$ is $1$-regular or if $M$ is the uniqueperfect matching of $G(M)$, respectively. Let $\nu_s(G)$ and $\nu_{ur}(G)$denote the maximum cardinality of an induced and a uniquely restricted matchingin $G$. Golumbic, Hirst, and Lewenstein (Uniquely restricted matchings,Algorithmica 31 (2001) 139-154) posed the problem to characterize the graphs$G$ with $\nu_{ur}(G) = \nu_{s}(G)$. We prove that the corresponding decisionproblem is NP-hard, which suggests that a good characterization is unlikely tobe possible.

On the hardness of deciding the equality of the induced and the uniquely restricted matching number