  To a $2\times2$ matrix $G$ with complex entries, we relate the sequence of
Laurent polynomial $L_n(z,G)=\tr \big(G\big[\begin{smallmatrix}z&0\\
0&z^{-1}\end{smallmatrix}\big]G^{\ast}\big)^n$. It turns out that for each
\(n\), the family $\big\{L_n(z,G)\big\}_G$, where $G$ runs over the set of all
$2\times2$ matrices, is a three-parametric family. A natural parametrization of
this family is found. The polynomial $L_n(z,G)$ is expressed in terms of these
parameters and the Chebyshev polynomial $T_n$. The zero set of the polynomial
$L_n(z,G)$ is described.
