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In this paper, first we consider some partitioned matrices having some special kind of blocks, and we obtain their eigenvalues and eigenvectors. From these, we deduce several results on the eigenvalues of partitioned matrices in the literature. Next we consider an extension of a generalized join of graphs introduced by Hedetniemi (On classes of graphs defined by special cutsets of lines. In: Many facets of graph theory. Proceedings of the conference held at Western MiGammagan University, Kalamazoo/Mi, 1968. Lecture notes in mathematics, vol 110, pp 171–189, 1969), we call it the M-join of Hk. The spectra of partitioned matrices allow to detect the adjacency, the Laplacian and the signless Laplacian spectrum of old and new type of join of graphs. Further, we have constructed several pairs of simultaneously A-cospectral, L-cospectral and Q-cospectral graphs.