Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

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The categories of this paper are: gr-qc
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The categories of this paper are: nlin
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The categories of this paper are: q-alg
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The categories of this paper are: hep-ex
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The categories of this paper are: hep-lat
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: astro-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: patt-sol
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The categories of this paper are: ao-sci
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The categories of this paper are: funct-an
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The categories of this paper are: supr-con
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The categories of this paper are: eess
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The categories of this paper are: q-fin