structural:scholar_edition, quality:correction
Presenting as a matrix group

The group SL 2 (Z) {\displaystyle {\text{SL}}_{2}(\mathbb {Z})} can be generated by the two matrices

since

The projection SL 2 (Z) → PSL 2 (Z) {\displaystyle {\text{SL}}_{2}(\mathbb {Z})\to {\text{PSL}}_{2}(\mathbb {Z})} turns these matrices into generators of PSL 2 (Z) {\displaystyle {\text{PSL}}_{2}(\mathbb {Z})}, with relations similar to the group presentation.