The Fibonacci sequence of numbers appears in many surprising places. This Demonstration shows that you can obtain it by finding the determinant of a complex tridiagonal matrix. By changing just a single matrix element you can obtain the Lucas numbers instead.

The theory behind these particular matrices and others like them can be found in N. D. Cahill, J. R. D'Errico, D. A. Narayan, and J. Y. Narayan, "Fibonacci Determinants," The College Mathematics Journal, 33(3), 2002 pp. 221–225.