An efficient method based on the Faddeev-Leverrier algorithm combined with the Adomian decomposition method is devised to facilitate the stability analysis of multi-input multi-output control systems. In contrast to prior eigenvalue algorithms, our method affords all eigenvalues of the state matrix, either real or complex. Specifically, the calculation of the complex eigenvalues is made possible through special canonical forms, mainly involving square root operators, of the characteristic equation of the state matrix. Moreover, the proposed method does not require an initial guess, which is often a matter of concern since an inappropriate guess can cause failure in such available schemes. For the sake of illustration, a number of numerical examples, including chemical reaction processes, are also provided that demonstrate the efficiency of our new technique.