Proof of a matrix equality

Question

OK guys, intuitively it seems correct to write

$$ x' \Sigma^{-1} x \Sigma = x x' $$

Where $\Sigma$ is a positive definite, symmetric matrix. To be honest, I have to say that I'm stuck in proving this equality... Can anybody show me the proper trick to get equality above?

Answer 1

Let $\Sigma \in \mathbb{R}^{2 \times 2}$.

The size on the left hand size is $2 \times 2$ but the size on the right hand side is $1 \times 1$. Hence it is not true.

Edit: Let $\Sigma=I_2$ and $x=(1,1)^T$. The left hand side is a diagonal matrix but the right hand side is not.