Descriptively, a statically determinate structure can be defined as a structure where, if it is possible to find internal actions in equilibrium with external loads, those internal actions are unique. The structure has no possible states of self-stress, i.e. internal forces in equilibrium with zero external loads are not possible.
Mathematically, this requires a stiffness matrix to have full rank.
A statically indeterminate structure can only be analytically analyzed by including further information like material properties and deflections. Numerically, this can be achieved by using methods like matrix structural analyses and finite element analyses.