Insights into a new model of matrix block 3 × 3

The problem of finding the eigenvalues of operators matrices is quite pertinent in operator theory. Thus, it would be interesting to keep this notion within the framework of operator matrix block 3 × 3. The novelty of our paper is to develop a new abstract setting in order to summarize a number of original results within the framework of coupled operator matrix block 3 × 3. Our approach exploits a rigorous study yielding a simple method for computation the eigenvalues of our above new considered model of matrix and provide a real progress in operator matrix theory and a positive answer of the recent work done by I. Marzouk et al. in ([17, New approach on the study of operator matrix, Georgian. Math. J., 31, 2,315-330, (2024)]). Furthermore, an example of neutron transport equation with partly elastic collision operator is presented to understand the validity and utility of our theoretical investigations.