In this paper, we study the total domatic partition problem for bipartite graphs, split graphs, and graphs with balanced adjacency matrices. We show that the total domatic partition problem is NP-complete for bipartite graphs and split graphs, and show that the total domatic partition problem is polynomial-time solvable for graphs with balanced adjacency matrices. Furthermore, we show that the total domatic partition problem is linear-time solvable for any chordal bipartite graph \( G \) if a \( \Gamma \)-free form of the adjacency matrix of \( G \) is given.