Some properties of Padovan matrices and bi-periodic Padovan matrices

Let ${\left(P_n\right)}_{n\ge 0}$ be the sequence of bi-periodic Padovan numbers and let ${\left(M_{p_n}\right)}_{n\ge 0}$ be the sequence of bi-periodic Padovan matrices. In this article we study when these matrices are diagonalizable and we obtain a certain connection with the Lucas number sequence. We also obtain some connections of these matrices with the generating matrix $Q$ for the Padovan numbers.