We describe the development of a multi-frequency electrical impedance tomography (EIT) system (KHU Mark1) with a single balanced current source and multiple voltmeters. It was primarily designed for imaging brain function with a flexible strategy for addressing electrodes and a frequency range from 10 Hz-500 kHz. The maximal number of voltmeters is 64, and all of them can simultaneously acquire and demodulate voltage signals. Each voltmeter measures a differential voltage between a pair of electrodes. All voltmeters are configured in a radially symmetric architecture in order to optimize the routing of wires and minimize cross-talk. We adopted several techniques from existing EIT systems including digital waveform generation, a Howland current generator with a generalized impedance converter (GIC), digital phase-sensitive demodulation and tri-axial cables. New features of the KHU Mark1 system include multiple GIC circuits to maximize the output impedance of the current source at multiple frequencies. The voltmeter employs contact impedance measurements, data overflow detection, spike noise rejection, automatic gain control and programmable data averaging. The KHU Mark1 system measures both in-phase and quadrature components of trans-impedances. By using a script file describing an operating mode, the system setup can be easily changed. The performance of the developed multi-frequency EIT system was evaluated in terms of a common-mode rejection ratio, signal-to-noise ratio, linearity error and reciprocity error. Time-difference and frequency-difference images of a saline phantom with a banana object are presented showing a frequency-dependent complex conductivity of the banana. Future design of a more innovative system is suggested including miniaturization and wireless techniques.