Powers and trace of symmetric powers of $2\times 2$ matrices and combinatorial, Fibonacci and Lucas identities

Abstract

Let $A$ be an arbitrary $2\times 2$ matrix. In \cite{Cisneros:PhD,Cisneros:I2x2M} I gave a formula for the trace of the $k$-th symmetric power of $A$ in terms of the anti-diagonal entries of $A^{k+1}$ and $A$. This was based on formulae that I found for the entries of the $k$-th power $A^k$ of the matrix $A$ in terms of its entries but I only sketched the idea of how I obtained such formulae. In this article I give the full proof of those formulae by counting some walks of length $k$ over the complete digraph of order $2$. I compare them with formulae for $A^k$ given by Mc Laughlin in \cite{McLaughlin:CIDnP2x2M} and by Williams in \cite{Williams:nthP2x2M}. This leads to combinatorial identities, in particular expressions for Fibonacci and Lucas numbers.