  Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be a diagonal
matrix of the degrees of $G$. In 2017, Nikiforov defined the
$A_{\alpha}$-matrix of $G$ as \begin{equation*} A_{\alpha}(G)=\alpha
G)+(1-\alpha)A(G), \end{equation*}d where $\alpha\in[0,1]$ is an arbitrary real
number. The largest eigenvalue of $A_{\alpha}(G)$ is called the
$A_{\alpha}$-spectral radius of $G$. Let $n$, $t$, $k$ be positive integers,
satisfying $t\geq1$, $k\geq2$, $n\geq k+2$, and $n\equiv k$ (mod $2$). In this
paper, for $\alpha\in[0,\frac{1}{2}]$, we determine the extremal graphs with
the maximum $A_{\alpha}$-spectral radius among all $t$-connected graphs on $n$
vertices with matching number $\frac{n-k}{2}$ at most. This generalizes some
results of O (2021) and Zhang (2022).
