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

Abstract

In this paper we give a new formula for the $n$-th power of a $2\times 2$ matrix. More precisely, we prove the following: Let $A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ be an arbitrary $2\times 2$ 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 $n\ge 1$, \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.

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