{"paper":{"title":"Combinatorial Identities Deriving From The $N$-th Power Of A $2\\Times 2$ Matrix","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"James Mc Laughlin","submitted_at":"2018-12-28T18:52:54Z","abstract_excerpt":"In 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.11168","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}