The purpose of this project is to investigate the automorphisms of what is arguably the simplest example of a free spectrahedron whose automorphisms have not yet been classified. A spectrahedron is the scalar solution set of a Linear Matrix Inequality (LMI), generalizing the notion of a polytope from linear programming. They are the basis for semidefinite programming (SDP) within convex optimization. Free spectrahedra are their matricial analogs, and have more structure and are hence more tractable. Free spectrahedra have connections to quantum information theory and systems engineering, and the study of their automorphisms has close ties to noncommutative algebra.