  This paper proves the $r \times r$ minors of an $n \times n$ symmetric matrix
of indeterminates are a tropical basis when $r = 2$, $r = 3$, or $r = n$, and
are not when $4 < r < n$ or $r = 4, n > 12$. In the process, it introduces two
new notions of rank for symmetric matrices coming from tropical geometry, the
symmetric tropical and the symmetric Kapranov rank, which are the symmetric
versions of their standard counterparts defined by Develin, Santos, and
Sturmfels.
