This paper presents a comprehensive solution for over-modulation of direct matrix converter using carrier-based control. The over-modulation operating region is classified, based on the choice of duty-ratios (input/output) and the maximum achievable output voltage amplitude, into three different modes named as Input-Side Over-modulation (ISO), Output-Side Over-modulation (OSO) and Simultaneous Input and Output-Side Over-modulation (SIOSO). The boundaries for operation of the matrix converter in each of the above three operating modes are established. The ISO is sub-divided into two modes based on the capability to control the power factor of input phase currents. The two regions are named as Input-Side Over-modulation with Power Factor Control (ISO with PFC) and Input-Side Over-modulation without Power Factor Control (ISO without PFC) respectively. A modulation algorithm for the choice of duty-ratios to allow a smooth transition between linear modulation and the ISO operating modes is proposed. A modification to the choice of the reference output voltage modulating signals for OSO that forces the converter into square wave mode operation with peak amplitude of the reference signal that is only twice the peak amplitude in linear modulation, is proposed. The SIOSO is used to obtain the maximum possible voltage transfer ratio of 105.3% from the matrix converter with over-modulation. The Total Harmonic Distortion (THD) of the output voltages and input currents are presented for all over-modulation operating modes. Theoretical calculations are verified with experiments on a laboratory prototype of the direct matrix converter.