  All matrices we consider have entries in a fixed algebraically closed field
$K$. A minor of a square matrix is principal means it is defined by the same
row and column indices. We study the ideal generated by size $t$ principal
minors of a generic matrix, and restrict our attention to locally closed
subsets of its vanishing set, given by matrices of a fixed rank. The main
result is a computation of the dimension of the locally closed set of $n\times
n$ rank $n-2$ matrices whose size $n-2$ principal minors vanish; this set has
dimension $n^2-n-4$.
