Institute of Mathematics of the Czech Academy of Sciences

In this note, we propose a research problem and two conjectures on the determinant of pentadiagonal matrices. At the same, we recall some results on pentadiagonal matrices and use them to provide new and short proofs of some recent determinantal identities for such matrices in the symmetric and skew-symmetric cases. This unified approach can be extended to further spectral consideration. In addition, two examples of explicit formulas of two non-trivial pentadiagonal matrices are provided.