In this study, we considered circulant matrices whose elements are Fibonacci polynomials. Then, we computed their determinants in two ways. In this content, we initially benefited from Chebyshev polynomials of the second kind. In the second way, we utilized some basic matrix operations. Moreover, we computed the inverse of these matrices in a general form. Furthermore, we found some kind of norms such as the Euclidean norm, upper and lower bounds for $\Vert {\mathcal{C}}_n{\Vert }_2$. In addition, we added some illustrative examples to make the results clear for the readers. In addition to these, we provide a MATLAB-R2023a code that writes the circulant matrix with the Fibonacci polynomial inputs, as well as computes Euclidean norms and bounds for their spectral norms.

In 1875, Glaisher systematically found several interesting determinant expressions of numbers, including Bernoulli, Cauchy, and Euler numbers. In this paper, we identify several determinants that express Euler polynomials. Goy and Shattuck presented several determinantal expressions of some families of Toeplitz–Hessenberg matrices with Tribonacci number entries. However, a determinant expression of Tribonacci numbers has not been studied much. By using a similar form of determinants to Euler's, we also give some determinant representations of generalized Tribonacci numbers.