Generalization of the Matrix Determinant Lemma and its application to the controllability of single input control systems

Linear control theory provides a rich source of inspiration and motivation for development in the matrix theory. Accordingly, in this paper, a generalization of Matrix Determinant Lemma to the finite sum of outer products of column vectors is derived and an alternative proof of one of the fundamental results in modern control theory of the linear time--invariant systems $\dot x=Ax+Bu,$ $y=Cx$ is given, namely that the state controllability is unaffected by state feedback, and even more specifically, that for the controllability matrices $\mathcal{C}$ of the single input open and closed loops the equality $\det\left(\mathcal{C}_{(A,B,C)}\right)$ $=\det\left(\mathcal{C}_{(A-BK,B,C)}\right)$ holds.