  Motivated by some recent developments in abstract theories of quadratic
forms, we start to develop in this work an expansion of Linear Algebra to
multivalued structures (a multialgebraic structure is essentially an algebraic
structure but endowed with some multivalued operations).
  We introduce and study matrices and determinants over a commutative
superrings (roughly, a ring where the sum and product are multivalued) and
study linear systems and vector spaces over superfields. As an application, we
obtain a fundamental result to the development of a theory of algebraic
extensions of superfields.
