Given $A \in\mathbb{R^{7\times8}}$, $B \in\mathbb{R^{8\times8}}$, $C \in\mathbb{R^{8\times7}}$ and provided that B is invertible, how can one check whether $$rank(AC) = rank(ABC)$$ is true?
$B$ is invertible and $n\times n$, so it is linearly equivalent to $I_n$, but I am not sure if this is what we need from the fact that $B$ is invertible.
$$\pmatrix{1&2\cr}\pmatrix{0&1\cr1&0\cr}\pmatrix{1\cr-2\cr}=0$$
$$\pmatrix{1&2\cr}\pmatrix{1\cr-2\cr}\ne0$$