Writing the scalar product using a determinant

Question

Let $A \in \mathbb{R}^{n \times n}$ be symmetrical and positive definite. Does the following statement hold true for $x \in \mathbb{R}^n$?

$$\det(x^TAx) = \det(x^TxA)$$

And if so, how can it be proven? (Or disproven?)

Answer 1

It is not true. Take $x = \pmatrix{1\\0}$ and $A = \pmatrix{2&0\\0&2}$, for example.