<p>With ubiquitous growth and evolution of data in real life applications, processing and analyzing dynamic data has significant potential to drive problem-solving across various domains. Incremental three-way approximations methods have garnered increasing concerns due to their ability to effectively enhance knowledge maintenance efficiency in dynamic data environments. In this study, we propose incremental updating methods for three-way approximations within incomplete neighborhood multi-granularity rough sets (INMGRSs) under the scenarios of addition or deletion of attributes. We devise a novel matrix representation for the three-way approximations in INMGRSs with the assistance of matrix operations. By analyzing the changes in relevant matrices resulting from attribute variations across multi-granularity, we develop and analyze the matrix-based mechanisms for updating these matrices, reducing the effort of unnecessary re-computation. In particular, we present two incremental algorithms in according with these mechanisms. Through a comprehensive comparative analysis, we conduct experimental verification to assess the performance of the proposed algorithms against existing incremental algorithms.</p>