# What can we say about the product of minors of a symmetric matrix?

I've been working on the following question with little succes for the past few days and could use some help of a kind stranger

For a symmetric matrix $$X$$ with the $$k$$-th principal minor of $$X$$ noted as $$X_k$$ (so consisting of the first $$k$$ rows and columns of $$X$$). Suppose $$X$$ is such that $$\det(X_k)>0 \forall k\in [n-1]$$, but $$\det(X)<0$$. Define $$a_i = (-1)^i M_{(i,n)}$$ where $$M_{(i,n)} = \det(X_{(i,n)})$$ where $$X_{(i,n)}$$ is $$X$$ without the $$i$$-th row and $$n$$-th column. I wnat to show $$a^Ta = \det(X)\det(X_{n-1})$$

I've already shown that $$\det(X)\det(X_{n-1}) = a_n\big(\sum_{i=1}^{n}aix_{n,i}\big)$$ (with $$x_{ij}$$ just the $$i,j$$-th entry of $$X$$) and that $$X_k$$ is positive definite for all $$k=1,\dots n-1$$ because of Sylvesters criterion.

I've been trying to write out $$a^TXa$$ for hours but i havent come further than $$a^TXa > 2\det(X)\det(X_{n-1}) - a_n^2x_{n,n}$$ but thats a little useless since you don't want to strictly bound it... I was wondering what I could say about $$x_{n,n}$$ knowing all that stuff about the determinants and stuff.

What do I do?