WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 12, 2013
On the Explicit Determinants and Singularities of r-circulant and Left r-circulant Matrices with Some Famous Numbers
Authors: , ,
Abstract: Let A be a r-circulant matrix and B be a left r-circulant matrix whose first rows are (P1, P2, ... , Pn), (Q1, Q2, ... , Qn), (J1, J2, ... , Jn) and (j1, j2, ... , jn) respectively, where Pn is the Pell number, Qn is the Pell- Lucas number, Jn is the Jacobsthal number and jn is the Jacobsthal-Lucas number. In this paper, by using the inverse factorization of polynomial of degree n, the explicit determinants of A and B whose first rows are (P1, P2, ... , Pn) and (Q1, Q2, ... , Qn) are expressed by utilizing only Pell numbers, Pell-Lucas numbers and the parameter r, and the explicit determinants of A and B whose first rows are (J1, J2, ... , Jn) and (j1, j2, ... , jn) are expressed by utilizing only Jacobsthal numbers, Jacobsthal-Lucas numbers and the parameter r. The results not only extend the original results, but also simpler in forms. Also, the singularities of those matrices are discussed. Furthermore, four identities of those famous numbers are given.
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Keywords: r-circulant matrix, Left r-circulant matrix, Determinant, Singularity, Pell numbers, Pell-Lucas numbers, Jacobsthal numbers, Jacobsthal-Lucas numbers