If $\displaystyle A\in M_{n}(\mathbb{F})$,where$\displaystyle \mathbb{F}=\mathbb{R}$ or $\displaystyle \mathbb{F}=\mathbb{C} $
and $\displaystyle A$ satisfy:$\displaystyle A'=A$ (where$\displaystyle A'$ is the transposed matrix of $\displaystyle A$).we say:
$\displaystyle A$ is a symmetric matrix

ok, If $\displaystyle A=\begin{bmatrix}a_{11}& ... & a_{1n}\\ ... & ... & ...\\ a_{n1}& ... & a_{nn}\end{bmatrix}. $

we denote the principal minor of $\displaystyle A$ of order $\displaystyle r$ by $\displaystyle det\begin{bmatrix}a_{i_{1}i_{1}}& ... & a_{i_{1}i_{r}}\\ ... & ... & ...\\ a_{i_{r}i_{1}}& ... & a_{i_{r}i_{r}}\end{bmatrix}.$
where$\displaystyle 1\leq i_{1}< ... < i_{r}\leq n$

and my question is: If $\displaystyle A$ is a symmetric matrix or a Hermite Matrix, then $\displaystyle A$ has a nonzero principal minor which order equal $\displaystyle rank(A)$

another question is: the Geometric meaning of Symmetric matrix is?