Application of metric tensor on a second-rank tensor

Question

I was just wondering if the below statement is true:

$$g_{\mu\nu} T^{\mu\rho} = T_\nu^\rho$$

Meaning whether or not the metric tensor has the same effect on all tensors that it has on vectors.

Answer 1

If you change it to $T_\nu{}^\rho$ (so that it's clear which of the two indices is first), then yes. Raising and lowering indices this way can be done for any tensors, not just vectors.