If $v$ is a random vector and $A$ is a random matrix with $E[A] = C$, is it true that $E[v^TAv] = E[v^TCv]$?

Question

If $v$ is a random $n$-dimensional vector and $A$ is a random $n$-by-$n$ matrix independent of $v$ with $\mathbb E[A] = C$, is it true that $\mathbb E[v^TAv] = \mathbb E[v^TCv]$ ?

Answer 1

True if $A$ is independent of $v$ not true in general. Counter-example in one dimension: Let $v \sim N(0,1)$ and $A=v^{2}$.