Invertible product of different-dimensional matrices

Question

We have the following situation: $A$ is an $n\times m$ matrix, $B$ is an $m\times n$ matrix and $C$ is some invertible $n\times n$ matrix. Can we, in general, say $$A(BCA)^{-1}B=C^{-1}?$$ Clearly, if $m=n$ and if $A$ and $B$ are invertible, then this is true. But what if $n\not=m$? Is this then true or false? Under what conditions is it true? Thanks for your help!!

Answer 1

In genera, $\operatorname{rank}(XY)\le\min\{\operatorname{rank}(X),\operatorname{rank}(Y)\}$.

If $m>n$, the ranks of $X=B$ and in turn $BCA=B(CA)$ are at most $n$ and hence the latter is not invertible in the first place.

If $n>m$, the ranks of $X=A$ and in turn $A(BCA)^{-1}B=A\left[(BCA)^{-1}B\right]$ are at most $m$ and hence the latter is not invertible.

So, in order that equality holds, $m$ must be equal to $n$, but this case is trivial.