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This paper deals with the problem of recovering the output signals from linear systems with unknown model order. Because the linear system identification problem with unknown model order can be formulated as the matrix rank minimization problem by applying the subspace identification method, this paper takes a matrix rank minimization approach to the signal recovery problem and formulates the problem as the mixed matrix rank and the Frobenius norm minimization problem. In order to solve this problem approximately, this paper proposes the null space based alternating optimization (NSAO) identification algorithm. Modifying the NSAO algorithm, this paper proposes a sequential identification algorithm, which enables us to identify the model order and the true output of switched linear systems. Experimental results using the output signals from the Kinect sensor show that the proposed algorithm can identify the model order and the true output and that the sequential algorithm can recover the missing signals generated from switched linear systems efficiently.