  A minor is principal means it is defined by the same row and column indices.
Let $X$ be a square generic matrix, $K[X]$ the polynomial ring in entries of
$X$, over an algebraically closed field, $K$. For fixed $t\leq n$, let
$\mathfrak P_t$ denote the ideal generated by the size $t$ principal minors of
$X$. When $t=2$ the resulting quotient ring $K[X]/\mathfrak P_2$ is a normal
complete intersection domain. When $t>2$ we break the problem into cases
depending on a fixed rank, $r$, of $X$. We show when $r=n$ for any $t$, the
respective images of $\mathfrak P_t$ and $\mathfrak P_{n-t}$ in the localized
polynomial ring, where we invert $\det X$, are isomorphic. From that we show
the algebraic set given by $\mathfrak P_{n-1}$ has a codimension $n$ component,
plus a codimension 4 component defined by the determinantal ideal (which is
given by all the submaximal minors of $X$). When $n=4$ the two components are
linked, and we prove some consequences.
