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