The applications of generalized inverse systematic non-square binary matrices span many domains including mathematics, error-correction coding, machine learning, data storage, navigation signals, and cryptography. In particular, they are employed in the McEliece and Niederreiter public key cryptosystems. For a systematic non-square matrix H of size (n-k) x n, n > k, there exist 2^{k x (n-k)} distinct inverse matrices. This paper presents an algorithm to generate these matrices as well as a method to construct a random inverse for systematic and non-systematic binary matrices. The proposed approach is shown to have lower computational complexity than the well-known Gauss-Jordan techniques. The application to public key cryptography (PKC) is also discussed.