From matrix notation to element-wise

Question

Let $B\in\mathbb R^{q\times p}, C\in\mathbb R^{q\times q}$ be two real matrices, and $C$ invertible and symmetric.

Define $A= B^\top C^{-1} B$.

Can we express the elements of $A$, $a_{i,j}$, using elements of $B$ and $C$ in general ?

Answer 1

@MostafaAyaz's objection remains, even with the constraints $\det C\ne0,\,C^\top=C$. You can write$$A_{ij}=B^\top_{ik}C^{-1}_{kl}B_{lj}=B_{ki}C^{-1}_{kl}B_{lj},$$but in general you have to do it with $C^{-1}$, not $C$. Even the special case $p=q,\,B=I_p$ requires you to somehow write components of $C^{-1}$ in terms of $C$, which you can't do, except verbosely. You certainly can't write something nice like $A_{ij}=B_{ki}B_{lj}X_{klmn}C_{mn}$.