This chapter is on elementary matrix operations using a state-space or, equivalently, quasi-separable representation. It is a straightforward but unavoidable chapter. It shows how the recursive structure of the state-space representations is exploited to make matrix addition, multiplication and elementary inversion numerically efficient. The notions of outer operator and inner operator are introduced as basic types of matrices playing a central role in various specific matrix decompositions and factorizations to be treated in further chapters.