Explicit inverses of some special matrices and their applications

As powerful mathematical structures, matrices have widespread use in real-world applications in various disciplines such as engineering, physics, computer science, and economics. In recent years, the topic of matrix inverses has started to attract the attention of many researchers. In this study, we investigate the inverse of some lower triangular polynomial matrices, which are formed by conditional polynomial sequences, with the help of some analytical techniques. We derive some correlations between conditional polynomial matrices and the T-nomial matrices of the first and of the second kind. In addition, we get factorizations of the conditional polynomial matrices via T-nomial matrices. Moreover, we obtain several combinatorial identities and provide more generalized results. Finally, we provide some numerical results that explain our method is faster and more efficient than MATHEMATICA?s Inverse method while computing the inverses of non-singular conditional polynomial matrices.