  Let $n,m$ be integers such that $1\leq m\leq (n-2)/2$ and let
$[n]=\{1,\ldots,n\}$. Let $\mathcal{G}=\{G_1,\ldots,G_{m+1}\}$ be a family of
graphs on the same vertex set $[n]$. In this paper, we prove that if for any
$i\in [m+1]$, the spectral radius of $G_i$ is not less than
$\max\{2m,\frac{1}{2}(m-1+\sqrt{(m-1)^2+4m(n-m)})\}$, then $\mathcal{G}$ admits
a rainbow matching, i.e. a choice of disjoint edges $e_i\in G_i$, unless
$G_1=G_2=\ldots=G_{m+1}$ and $G_1\in \{K_{2m+1}\cup (n-2m-1)K_1, K_m\vee
(n-m)K_1\}$.
