Invertibility of a Particular Matrix Product

Question

Let $\boldsymbol{A}$ be an $n \times k$ matrix, $\boldsymbol{B}$ be an $n \times n$ matrix and $\boldsymbol{C}$ be an $n \times k$ matrix with $k < n$. If it is known that $\boldsymbol{A}^{\top} \boldsymbol{B} \boldsymbol{C}$ is invertible and also that $\boldsymbol{B}$ is invertible, does this imply $\boldsymbol{A}^{\top}\boldsymbol{C}$ is invertible?

Answer 1

$A^tB$ is not even a square matrix, so it can't be invertible...

Edit: for $A^tC$ it's also not true. Take n=2, k=1 and the matrices $A=\begin{pmatrix}1\\0\\\end{pmatrix}$, $B=\begin{pmatrix}0&1\\1&0\\\end{pmatrix}$, $C=\begin{pmatrix}0\\1\\\end{pmatrix}$.