Metric tensor derivative identity

Question

In this paper, the following identity is used without proof nor reference:

$$g^{\alpha \mu} g^{\beta \lambda} \dot{g}_{\alpha \beta} \equiv -\dot{g}^{\mu \lambda}$$

where $g$ is metric tensor and $\dot{g}$ is derivative of $g$.

How can this identity derived?

Answer 1

I found out that this is from the matrix identity, $$\partial X^{-1} = - X^{-1} (\partial X) X^{-1}$$

Proof for this identity is in here.

With this, we have $$X (\partial X^{-1}) X = - \partial X$$

By substituting $X$ with $g^{\mu \lambda}$, we get the identity!