  For a set of graphs $\mathcal{F}$, let $\ex(n,\mathcal{F})$ and
$\spex(n,\mathcal{F})$ denote the maximum number of edges and the maximum
spectral radius of an $n$-vertex $\mathcal{F}$-free graph, respectively.
Nikiforov ({\em LAA}, 2007) gave the spectral version of the Tur\'an Theorem by
showing that $\spex(n, K_{k+1})=\lambda (T_{k}(n))$, where $T_k(n)$ is the
$k$-partite Tur\'an graph on $n$ vertices. In the same year, Feng, Yu and Zhang
({\em LAA}) determined the exact value of $\spex(n, M_{s+1})$, where $M_{s+1}$
is a matching with $s+1$ edges. Recently, Alon and Frankl~(arXiv2210.15076)
gave the exact value of $\ex(n,\{K_{k+1},M_{s+1}\})$. In this article, we give
the spectral version of the result of Alon and Frankl by determining the exact
value of $\spex(n,\{K_{k+1},M_{s+1}\})$ when $n$ is large.
