<p>Matrix-product (MP) codes are a class of long classical codes formed by combining multiple commensurate classical codes with a defining matrix. MP codes with certain special defining matrices provide a feasible approach for constructing entanglement-assisted quantum error-correcting codes (EAQECCs) with favorable parameters. This paper focuses on constructing various EAQECCs using MP codes with non-singular by columns (NSC) quasi-orthogonal matrices as defining matrices. To achieve this, we propose two general methods for constructing NSC quasi-orthogonal matrices. Additionally, to further facilitate the search for such matrices, we present two effective algorithms. Based on these matrix construction methods, we propose a general method and two specific constructions for deriving EAQECCs from MP codes. By applying the propagation rules, we construct various EAQECCs with flexible parameters. Furthermore, we present an improved version for generating various EAQECCs from MP codes, which utilizes the concept of relative Euclidean hulls of classical codes. Numerical examples, comparisons and further remarks are provided to illustrate these results.</p>