Abstract. Many mathematicians investigated in papers various types of integer matrices the entries of which satisfy a second order recurrence. Some of the authors used methods leading to obtain real or complex factorizations of the Fibonacci or the Lucas numbers. Civciv (2008) computed the determinant of a five-diagonal matrix with the Fibonacci numbers as its entries. His result is given more generally and completely in this paper. It is showed that the determinant of a matrix, the entries of which are the Gibonacci numbers, is related to the values of the Fibonacci polynomial. Calculations are done by using the eigenvalues of the given matrix.

Received: February 20, 2013

AMS Subject Classification: 11B39, 11C20, 15B36

Key Words and Phrases: generalized Fibonacci number, determinant, five-diagonal matrix, eigenvalue, Fibonacci polynomial

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