For a generic set M of 3x3 matrices over C we find necessary and sufficient conditions when A4 is simultaneously self-adjoint.
Moreover, for a set of complex hermitian matrices we can tell if there
exists a linear combination of matrices which is positive definite. Every
M can be identified with a determinantal representation of a cubic hypersurface. This allows us to use the tools of algebraic geometry. The question of definiteness can be solved by using semidefinite programming.