inner product and multiply matrices

Question

Please could you give me any idea to solve this question I do not know how to start to solve this. thanks in advance.

Let $A$ and $B$ be $n$-square complex matrices. If, for every $x \in \Bbb C^n$:

$(Ax, x) = (Bx, x)$,

Does it follow that $A = B$? What if $x ∈ \Bbb R^n$?

Answer 1

No. Set

$$A= \begin{bmatrix}0 & -1\\ 1 & 0\end{bmatrix}$$ and $B=0$, for example.

Answer 2

Try to solve the problem for $n=2$ (for $n=1$ the property trivially holds). Is it true that $$((A-B)x,x)=\begin{bmatrix} x_1 &x_2\end{bmatrix}\begin{bmatrix} a & b\\ c & d\end{bmatrix} \begin{bmatrix} x_1\\ x_2\end{bmatrix}=ax_1^2+(b+c)x_1x_2+dx_2^2=0,\;\; \forall x_1,x_2\in\mathbb{R}$$ implies that $a=b=c=d=0$?

By taking $x_1=1$ and $x_2=0$ we obtain that $a=0$. Similarly for $x_1=0$ and $x_2=1$ we get $d=0$. So does $(b+c)x_1x_2=0$ imply that $b=c=0$?

What may we conclude? What happens when $n>2$?