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Linear control theory provides a rich source of inspiration and motivationfor development in the matrix theory. Accordingly, in this paper, ageneralization of Matrix Determinant Lemma to the finite sum of outer productsof column vectors is derived and an alternative proof of one of the fundamentalresults in modern control theory of the linear time--invariant systems $\dotx=Ax+Bu,$ $y=Cx$ is given, namely that the state controllability is unaffectedby state feedback, and even more specifically, that for the controllabilitymatrices $\mathcal{C}$ 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.

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