The problems of roots existence for different classes of matrix such as simple and semisimple matrixes from $SL_2({\mathbb{F}})$, $ SL_2({\mathbb{Z}})$ and $ GL_2({\mathbb{F}})$ are solved.
First time, we introduce the concept of extended special linear group $ESL_2(\mathbb{F})$, which is generalization of the matrix group $SL_2(\mathbb{F})$, where $\mathbb{F}$ is arbitrary perfect field.
The group of unimodular matrices are generalized by us, their structures are found.