The input and output matrix mapsBandCof a linear multivariable systemS(A, B, C)play an important role in determining the behaviour of the system. Square systems with the productCBfull rank possess a simple state-space geometry which is deployed in the present paper for the derivation of an explicit state-space characterization of the inverse system. An efficient algorithm for the inversion of a system is then obtained. The elegance of the results discussed enables the examination of the duality between poles/modes and zeros/zero-directions. Finally, an extension of the above to systems withCBrank-deficient is undertaken.