Abstract: This paper studies the spectral Turán problems for simultaneously prohibiting the matching M_s+1 of size s+1 and the chromatic critical graph F with chromatic number r+1 . By introducing combinatorial parameters such as the covering number and independent covering number of the graph, we establish a theoretical framework for analyzing the spectral extremal values of degenerate graph families. Under this framework, we prove that when s is sufficiently large, the maximum spectral radius of an n -order \M_s+1,F\\text-free graph reaches the complete r -partite graph G(n,r,s) , and this extremal graph is unique. This result not only extends the results of Alon and Frankl on edge extremal values to the spectral case, but also provides a complete solution to the spectral extremal value problem under the constraint of the coexistence of matchings and chromatic critical graphs.