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FIBONACCI MATRICES

In this chapter, we will take advantage of the power of matrices to extract new identities and results involving Fibonacci and Lucas numbers; see the Appendix for a short discussion of matrices.

20.1 THE Q-MATRIX

First we will demonstrate a close link between matrices and Fibonacci numbers. To this end, consider the matrix

equation

In 1960, Charles H. King studied this matrix, called the Q-matrix, for his Master's Thesis at then San Jose State College, California [360]. Notice that c020-math-001, where c020-math-002 denotes the determinant of the square matrix M. Then

equation

Similarly, c020-math-003.

Clearly, an interesting pattern is emerging. More generally, we have the following intriguing result. We will establish it using PMI.

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