Combinatorial Identities Deriving From The $N$-th Power Of A $2\Times 2$ Matrix

In this paper we give a new formula for the $n$-th power of a $2\times2$ matrix. More precisely, we prove the following: Let $A= \left ( \begin{matrix} a & b \\ c & d \end{matrix} \right )$ be an arbitrary $2\times2$ matrix, $T=a+d$ its trace, $D= ad-bc$ its determinant and define $$y_{n} :\,= \sum_{i=0}^{\lfloor n/2 \rfloor}\binom{n-i}{i}T^{n-2 i}(-D)^{i}.$$ Then, for $n \geq 1$, \begin{equation*} A^{n}=\left ( \begin{matrix} y_{n}-d \,y_{n-1} & b \,y_{n-1} \\ c\, y_{n-1}& y_{n}-a\, y_{n-1} \end{matrix} \right ). \end{equation*} We use this formula together with an existing formula for the $n$-th power of a matrix, various matrix identities, formulae for the $n$-th power of particular matrices, etc, to derive various combinatorial identities.