We consider a proximal point method for solving the nuclear norm regularized matrix least squares problem with equality and inequality constraints. We show that the soft thresholding operator is strongly semismooth everywhere. For the inner subproblems, due to the presence of inequality constraints, we reformulate the problem as a system of semismooth equations. Then an inexact smoothing Newton method is proposed to solve this reformulated semismooth system. At each iteration, we apply the BiCGStab iterative solver to obtain an approximate solution to the generated smoothing Newton linear system. Numerical experiments on a variety of large scale matrix least squares problems, where the matrices involved have some special structures, show that the proposed proximal point method is very efficient.