Let $A$ be a symmetric matrix with rank $r$. Is it true that there exists at least one non-zero $r\times r$ leading minor .
Here, a $r\times r$ leading minor is the determinate of a submatrix of $A$ with rows and columns from $i_1,i_2,\cdots,i_r$.
It is easy that if $A$ is a real matrix, then it is right....However, for other fields, I do not whether this is right...