Product of a specific $(0,2)$ and $(2,0)$ tensor (Minkowski Metric tensor)

Question

How to calculate $$\eta_{\mu\nu}\eta^{\mu\nu}$$ Where $$\eta=\begin{bmatrix} -1 \\ &1 \\&&1\\&&&1\end{bmatrix}$$ All other entries are $0$.

Answer 1

Writing $\eta^{\mu\nu}$ as $$\eta^{\mu\nu}=\eta^{\mu\alpha}\eta^{\nu\beta}\eta_{\alpha\beta}$$ Thus $$\eta_{\mu\nu}\eta^{\mu\nu}=\eta_{\mu\nu}\eta^{\mu\alpha}\eta^{\nu\beta}\eta_{\alpha\beta}$$ Rearranging $$\eta_{\mu\nu}\eta^{\mu\nu}=\eta^{\mu\alpha}\eta_{\alpha\beta}\eta^{\nu\beta}\eta_{\mu\nu} \\ =\eta^\mu_\beta \eta^\beta_\mu=4$$ Or equivalently since $$\eta^{\mu\nu}\eta_{\mu\chi}={\delta^\nu}_\chi\,$$ Where $\delta$ is Kronecker delta.

Substituting $\chi=\nu$ we get 4. Note that this only works because $\eta$ is symmetrical.