Summary: The matrix completion problem is to complete an unknown matrix from a small number of entries, and it captures many applications in diversified areas. Recently, it was shown that completing a low-rank matrix can be successfully accomplished by solving its convex relaxation problem using the nuclear norm. This paper shows that the alternating direction method (ADM) is applicable for completing a low-rank matrix including the noiseless case, the noisy case and the positive semidefinite case. The ADM approach for the matrix completion problem is easily implementable and very efficient. Numerical comparisons of the ADM approach with some state-of-the-art methods are reported.