I got to a point within an exercise, where I would like the following matrix to be invertible: $$ \mathbf{AC}^{-1}\mathbf{A}^T,$$ where $\mathbf{C}$ is a regular matrix and $\mathbf{A}$ has linearly independent rows (i. e. $\mathbf{AA}^T$ is regular).
The solution contains this matrix, but I don't see why it holds.
Since $A$ has linearly independent rows, it is impossible to find a non-zero vector $v$ such that $A^Tv=0$, since $AA^T$ is invertible. By defining $w=A^Tv$ we obtain $$v^TAC^{-1}A^Tv=w^TC^{-1}w$$ which is again non-zero since $C^{-1}$ is invertible. Hence the result.