On Factorization and Calculation of Determinant of Block Matrices with Triangular Submatrices

In this paper, we consider some block matrices of dimension $nm\times{nm}$ whose components are triangular matrices of dimension $n\times{n}$. We prove that the determinant of such block matrices is determined only by the diagonal elements of their submatrices and that this determinant is expressed as the multiplication of some subdeterminants. If the components of dimension $n\times{n}$ are all diagonal matrices, then we prove that such a block matrix can be written as a product of simpler matrices. Besides, we investigate the eigenvalues, the adjoint, and the inverse of such block matrices.