Let $\mathscr{G}_{n,β}$ be the set of graphs of order $n$ with given matching number $β$. Let $D(G)$ be the diagonal matrix of the degrees of the graph $G$ and $A(G)$ be the adjacency matrix of the graph $G$. The largest eigenvalue of the nonnegative matrix $A_α(G)=αD(G)+A(G)$ is called the $α$-spectral radius of $G$. The graphs with maximal $α$-spectral radius in $\mathscr{G}_{n,β}$ are completely characterized in this paper. In this way we provide a general framework to attack the problem of extremal spectral radius in $\mathscr{G}_{n,β}$. More precisely, we generalize the known results on the maximal adjacency spectral radius in $\mathscr{G}_{n,β}$ and the signless Laplacian spectral radius.