Fibonacci matrices and matrix representation of Fibonacci numbers (Q1964722)

Let \(a_{11}= a_{12}= a_{21}= 1\), \(a_{22}= 0\). It is well-known that the matrix \({\mathbf A}= [a_{ij}]\), \(i,j= 1,2\), and its integral powers \({\mathbf A}^n\) satisfy the Fibonacci-type relation \({\mathbf A}^{n+1}= {\mathbf A}^n+ {\mathbf A}^{n-1}\), where \({\mathbf A}^0= \mathbf{1}\), the identity matrix. They are called the Fibonacci matrices, and for \(n>0\), the entries \(a_{ij}^n\) of \({\mathbf A}^n\) are given by \(a_{11}^n= F_{n+1}\), \(a_{12}^n= a_{21}^n= F_n\) and \(a_{22}^n= F_{n-1}\). Using elementary matrix identities, the author rediscovers several known Fibonacci identities and finds a few new ones.