Is the inverse of the metric tensor also a tensor?

Question

In my textbook the inverse of the metric tensor ($g^{ij}$) is defined the following way: $$g^{ij}g_{jk}=\delta^i_k$$ Then if we have the matrices forward and backward transforamtion $A$ and $B$, we know that the metric tensor transforms like: $$g_{i'j'}=A^i_{\text{ } i'}A^j_{\text{ } j'}g_{ij}$$ How do we prove that the inverse of the metric also transforms like a tensor: $$g^{i'j'}=B^{i'}_{\text{ }i}B^{j'}_j g^{ij}$$ I tried transforming the first equation but got nonsense.