We establish an analogue of the classical Polya–Vinogradov inequality for 𝐺𝐿(2,𝔽𝑝)$GL(2,{\mathbb{F}}_p)$, where p is a prime. In the process, we compute the ‘singular’ Gauss sums for 𝐺𝐿(2,𝔽𝑝)$GL(2,{\mathbb{F}}_p)$. As an application, we show that the collection of elements in 𝐺𝐿(2,ℤ)$GL(2,\mathbb{Z})$ whose reduction modulo p are of maximal order in 𝐺𝐿(2,𝔽𝑝)$GL(2,{\mathbb{F}}_p)$ and whose matrix entries are bounded by x has the expected size as soon as 𝑥≫𝑝1/2+ε$x\gg p^{1/2+\text{ε}}$ for any ε>0$\text{ε}>0$.