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To a $2\times2$ matrix $G$ with complex entries, we relate the sequence ofLaurent 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 ofthis family is found. The polynomial $L_n(z,G)$ is expressed in terms of theseparameters and the Chebyshev polynomial $T_n$. The zero set of the polynomial$L_n(z,G)$ is described.

On a family of Laurent polynomials generated by 2x2 matrices