if $x^T*A*x=0$ can it be proven that det(A)=0

Question

Sorry, for my lack of knowledge on the subject but can someone please tell me if the statement is true or not

   if $x^T*A*x=0$ can it be proven that, det(A)=0

Answer 1

Nope. It can't be shown that

$\det A = 0 \tag 1$

from

$x^TAx = 0; \tag 2$

take

$A = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \tag 3$

and

$x = (1, 1)^T; \tag 3$

then (2) binds but

$\det A = -1; \tag 4$

or take

$A = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \tag 5$

and again choose $x$ as in (3); then again (2) holds but now

$\det A = 1. \tag 6$

Note that with $A$ as in (5), (2) binds for every $x$!