Quasi-tridiagonal matrices frequently arise in diverse areas of electrical engineering and computer sciences, for example in power system analysis and control, computer vision, image and signal processing, and in parallel computing. Some special types of tridiagonal and quasi-tridiagonal matrices have attracted much attention over the last few years. We present a novel cost-efficient recursive algorithm for numerically evaluating the determinant of a quasi-tridiagonal matrix with order n in the current paper, whose computational cost is estimated at 9n + O(1) flops. The algorithm is based on a specialised block diagonalization and a certain type of matrix factorization. Furthermore, an efficient way of evaluating the determinants of quasi-anti-tridiagonal matrices, without imposing any restrictive assumptions is also discussed. We provide some numerical results with simulations in MATLAB implementation to show the efficiency and accuracy of the proposed algorithm, and demonstrate its competitiveness with DETGTRI algorithm.