In the realm of linear algebra, the notion of matrix inversion plays a crucial role. While the inversion of square matrices is well-known and results in a unique inverse, however, the non-square inverse matrice is not unique and in fact, the number of inverses for a non-square matrix can be as vast as q^m(n−m), where q signifies the order of the underlying field. In this paper, we embark on a journey to construct these elusive inverse matrices, harnessing the power of arbitrary fields. Arbitrary fields, including prime fields, finite fields, real fields, and complex fields. These fields find practical applications that are essential to contemporary technology. I have written 5 MATLAB programs that able to construct random inverses in different fields based on the given algorithm.

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 2k 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.