Let $A$ be a $n \times n$ matrix, $u$ a $n \times 1$ matrix and $v$ a $1 \times n$ matrix. If $A$ and $(A+uv)$ are invertible, prove that $$ \det(A) \cdot v \, A^{-1} = \det(A+uv) \cdot v \, (A+uv)^{-1}. $$
I have numerical evidence that this is true, but I cannot prove it. Note that the term on the left does not depend on $u$!
Thank you.