The Baker-Akhiezer function theory was successfully developed in the middle of 70-th. This theory concerns of spectral theory and completely integrable nonlinear equations such as Korteweg de Vries equation, nonlinear Schrodinger equation, sine-Gordon equation, Kadomtsev-Petviashvili equation, etc. Later the theory was reproduced for the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchies. However, ... [Show full abstract] nothing is known about the Baker-Akhiezer (BA) function for the Maxwell-Bloch (MB) system or for the KarpmanKaup equations which contain prescribed weight functions characterizing inhomogeneous broadening of the main frequency. The main goal of the paper is to present a construction of matrix Baker-Akhiezer function associated with the Maxwell-Bloch equations. Using different Riemann-Hilbert problems on the complex plane with cuts we propose such a matrix function that has a unit determinant and takes an explicit form through the Cauchy integrals, hyperelliptic integrals and theta functions. This function solves the AKNS equations (the Lax pair for MB system) and generates a quasi-periodic finite-gap solution to the Maxwell-Bloch equations. The suggested function will be useful for studying of the long time asymptotic behavior of solutions of different initial-boundary value problems for the MB equations using the Deift-Zhou method of steepest descent and for an investigation of rogue waves of the Maxwell-Bloch equations.