Let G F ( q ) denote the finite field of order q = p e with p odd. Let M denote the ring of 2 × 2 matrices with entries in G F ( q ) . Let n denote a divisor of q − 1 and assume 2 ≤ n and 4 does not divide n . In this paper, we consider the problem of determining the number of n -th roots in M of a matrix B ∈ M . Also, as a related problem, we consider the problem of lifting the solutions of X 2 = B over Galois rings.