  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.
