In this paper, we consider the inverse optimal control problem for discrete-time Linear Quadratic Regulators (LQR), over finite-time horizons. Given observations of the optimal trajectories, or optimal control inputs, to a linear time-invariant system, the goal is to infer the parameters that define the quadratic cost function. The well-posedness of the inverse optimal control problem is first justified. In the noiseless case, when these observations are exact, we analyze the identifiability of the problem and provide sufficient conditions for uniqueness of the solution. In the noisy case, when the observations are corrupted by additive zero-mean noise, we formulate the problem as an optimization problem and prove that the solution to this problem is statistically consistent. The performance of the proposed method is illustrated through numerical examples.