If a symmetric bi-liear form is positive, then $a_{11}\cdot a_{nn}>a_{n1}\cdot a_{1n}$

Question

I need to prove that if a bi-linear form is symmetric and positive, then it's representative matrix $A=(a_{ij})_{i,j=1}^n$ satisfies: $$a_{11}\cdot a_{nn} > a_{n1}\cdot a_{1n}$$

I've tried for pretty long, without much success.

It seems that the last main minor (the determinant of the whole matrix) must be negative if the predicate isn't satisfied, but I can't figure out how to prove it.

Answer 1

The induced quadratic form, given a column vector $v$, is either $$ v^T A v $$ or half of that, depends on author. Anyway, what happens with row $$ v^T = (x,0,0,0,0, \ldots, 0,0,0,y)? $$