# ag.algebraic geometry – Symmetric matrices of Hyperbolic and elliptic type with certain kind of trace zero

I have been working on a problem related to determinantal varieties in Symmetric matrices. I am stuck at the following point and would like to get some reference/help for the following question.
Let $$mathbb{F}_q$$ be a finite field with odd characteristic and let $$S(2t, m)$$ be the set of all $$mtimes m$$ symmetric matrices over $$mathbb{F}_q$$ of rank $$2t$$(even). For some $$deltainmathbb{F}_q$$ a square (non-square) and $$kle m$$ let $$f^delta_k(X)= X_{11}+cdots+X_{k-1k-1}+delta X_{kk}$$. I want to know the cardinality of the following set
$${Ain S(2t, m): Atext{ is hyperbolic and }f^delta_k(A=0}.$$
Here, by hyperbolic $$A$$ we mean that the corresponding quadric $$XAX^T$$ is hyperbolic. I am stuck at this point. My approach was to use some induction on $$k$$. To do so, I was thinking to project a symmetric matrix to an $$m-1times m-1$$ matrix by deleting its first row and first column. But I have no control over the behavior of the fiber of this map. For example, if we take an $$m-1times m-1$$ symmetric matrix of rank $$2t-2, ;2t-1 or ;2t$$ and add a new row and column to obtain a matrix in $$S(2t, m)$$, what are the odds to get a hyperbolic matrix?

I know this stuff is quite classical and probably this problem is already well understood. But unfortunately, I could not find references that only gives the number of symmetric matrices that are hyperbolic and elliptic.