This manuscript proposes a generalized inverse for a dual matrix called dual Drazin generalized inverse (DDGI) which generalizes the notion of the dual group generalized inverse (DGGI). Under certain necessary and sufficient conditions, we establish the existence of the DDGI of a dual matrix of any index. Thereafter, we show that the DDGI is unique (whenever exists). The DDGI is then used to solve a linear dual system. We also establish reverse-order law and forward-order law for a particular form of the DGGI, dual Moore-Penrose generalized inverse (DMPGI), dual core generalized inverse (DCGI), and DDGI under certain suitable conditions. Finally, the partial-orders based on DCGI and DGGI are proposed.

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