Given $A = BAC$ and all matrices are invertible. What do we know about matrices $B$ and $C$?

Question

I know that the following equation holds

$A = BAC$,

and I also know that all matrices are invertible. Now, the equation is trivially true if $B$ and $C$ are the identity. What other $B$ and $C$ could satisfy the equation?

Answer 1

$$A = B\,A\,C \Leftrightarrow A\,C^{-1} = B\,A \Leftrightarrow A\,C^{-1}\,A^{-1} = B$$

Therefore, for ANY invertible matrices $A$ and $C$, the equation will be true if and only if $B = A\,C^{-1}\,A^{-1}$, which is invertible.

For a fixed known matrix $A$, the matrix $B$ can be considered a function of the matrix $C$, which is defined in the space of invertible matrices: $B(C) = A\,C^{-1}\,A^{-1}$.

---

Alternatively:

$$A = B\,A\,C \Leftrightarrow B^{-1}\,A = A\,C \Leftrightarrow A^{-1}\,B^{-1}\,A = C$$

Therefore, for ANY invertible matrices $A$ and $B$, the equation will be true if and only if $C = A^{-1}\,B^{-1}\,A$, which is invertible.

For a fixed known matrix $A$, the matrix $C$ can be considered a function of the matrix $B$, which is defined in the space of invertible matrices: $C(B) = A^{-1}\,B^{-1}\,A$.