I'm trying to understand tensors, but I've come across the following question:

The answers are hand written above. I'm finding it hard to understand these definitions. I really don't know what these tensors represent or what it means when both the indices are raised? Also are these standard definitions in special relativity or is it more likely that they are just made up by my professor for this assignment? I've always thought that things with two indices are matrices in general.

Thanks for any replies!

I'm trying to understand tensors, but I've come across the following question:

3. Let $T^{\mu\nu}$ by a $(2,0)$ tensor. Give the definitions of $T_\mu^{\,\nu}$, $T_{\mu\nu}$, and $T^{\mu}_{\,\nu}$. Then give explicit formulas for the following components in terms of components of $T^{\mu\nu}$: $T_{00},\,T_{10},\,T_{12},\,T_0^{\,1},\,T_1^{\,1},\,T_0^{\,0},\,T^1_{\,1}$
   $$T_\mu^{\,\nu}\equiv\eta_{\mu\sigma}T^{\sigma\nu} \\ T_{\mu\nu}\equiv\eta_{\sigma\mu}\eta_{\rho\nu}T^{\sigma\rho}\\ T^\mu_{\,\nu}\equiv\eta_{\sigma\nu}T^{\mu\sigma}$$

I'm finding it hard to understand these definitions. I really don't know what these tensors represent or what it means when both the indices are raised? Also are these standard definitions in special relativity or is it more likely that they are just made up by my professor for this assignment? I've always thought that things with two indices are matrices in general.