We point a generalized inverse $A_G$ of a singular square complex matrix $A$, with the property that the general solution of the equation $A^nx=0$ (and many other equations) can be expressed by means of $A_G$ for all positive integers $n$. This inverse is a solution of the system (5), also satisfied by the strong spectral inverse of Greville. Applications to various matrix equations and to linear algebraic systems are given.