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this paper we give a new formula for the $n$-th power of a $2\\times2$ matrix.\n  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,\n  $D= ad-bc$ 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*}\n  We use this formula together with an existing formula fo","authors_text":"James Mc Laughlin","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-12-28T18:52:54Z","title":"Combinatorial Identities Deriving From The $N$-th Power Of A $2\\Times 2$ 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