We propose a new systematic matrix approach, which is an alternative to the disordered virtual ground technique, the Kirchhoff’s law，and traditional Thevenin approaches, of designing the desired performance and simultaneously finding two Thevenin equivalent parameters for general complicated operational amplifier systems. One cannot apply the virtual ground method to non-ideal and partially ideal operational amplifier systems including instrumentation and frequency sensors. There has been no matrix method for efficiently solving a complicated operational amplifier circuit for conventional tedious approaches that apply Kirchhoff’s law. We also propose a new method of finding Thevenin equivalent circuits for general complicated operational amplifier systems that has significant merits over traditional approaches including the open-circuit voltage and short-circuit current methods, the matrix node voltage (mesh current) method, the modified nodal (mesh) analysis, the reference node r model, the reflected impedance method, and the virtual ground method. To compare the proposed and traditional approaches, one demonstrated example and one comparative example are chosen. Moreover, we report a Bessel low-pass filter that is mainly made up of an AD8618 quad op-amp. Its transfer function for existing methods is incorrect in the non-ideal case, and thus the important amplitude-frequency and phase-frequency characteristic curves will not be exact. An online electrocardiograph (ECG) device is an important tool for many industrial medicine applications and is investigated in this study to demonstrate the practical value of the proposed results.