Let $G$ be a simple graph with associated diagonal matrix of vertex degrees $D(G)$, adjacency matrix $A(G)$, Laplacian matrix $L(G)$ and signless Laplacian matrix $Q(G)$. Recently, Nikiforov proposed the family of matrices $A_α(G)$ defined for any real $α\in [0,1]$ as $A_α(G):=α\,D(G)+(1-α)\,A(G)$, and also mentioned that the matrices $A_α(G)$ can underpin a unified theory of $A(G)$ and $Q(G)$. Inspired from the above definition, we introduce the $B_α$-matrix of $G$, $B_α(G):=αA(G)+(1-α)L(G)$ for $α\in [0,1]$. Note that $ L(G)=B_0(G), D(G)=2B_{\frac{1}{2}}(G), Q(G)=3B_{\frac{2}{3}}(G), A(G)=B_1(G)$. In this article, we study several spectral properties of $ B_α$-matrices to unify the theories of adjacency, Laplacian, and signless Laplacian matrices of graphs. In particular, we prove that each eigenvalue of $ B_α(G) $ is continuous on $ α$. Using this, we characterize positive semidefinite $ B_α$-matrices in terms of $α$. As a consequence, we provide an upper bound of the independence number of $ G $. Besides, we establish some bounds for the largest and the smallest eigenvalues of $B_α(G)$. As a result, we obtain a bound for the chromatic number of $G$ and deduce several known results. In addition, we present a Sachs-type result for the characteristic polynomial of a $ B_α$-matrix.