tensor calculus - Lowering/raising metric indexes

So, I was chatting with a friend and we noticed something that might be very, very, very stupid, but I found it at least intriguing.

Consider Minkowski spacetime. The trace of a matrix $A$ can be written in terms of the Minkowski metric as $\eta^{\mu \nu} A_{\mu \nu} = \eta_{\mu \nu} A^{\mu \nu} = A^\mu_\mu$.

What about the trace of the metric? Notice that $\eta^\mu_\mu$ cannot be written as $\eta_{\mu \nu} \eta^{\mu \nu}$, because this is equal to $4$, not $-2$. It seemed to us that there is some kind of divine rule that says "You shall not lower nor raise indexes of the metric", because $\eta^{\mu \nu} \eta_{\nu \alpha} = \delta^\mu_\alpha \neq \eta^\mu_\alpha$. Is the metric immune to index manipulations? Is this a notation flaw or am I being ultra-dumb?

Answer

The mistake you made is this: $\eta^{\mu}_{\nu} \neq \eta_{\mu\nu} $. When you raise index $\mu$ from downstairs to upstairs, the matrix elements change. $\eta^{0}_{0} = 1$, $\eta_{00} = -1$. That is why if you take the trace of $\eta_{\mu\nu}$, you get 2, but if you take the trace of $\eta^{\mu}_{\nu}$ you get 4.