Proving the inverse of an elimination matrix?

Question

I am having some trouble proving that the inverse of $M_k = I - m_ke_k^T$ is equal to $L_k = I + m_ke_k^T$. Where I is an $n x n$ identity matrix, and $M_k$ is the elimination matrix for the $k^{th}$ column $a_k$ of A. I believe I should start by taking the inverse of $M_k$ to get $ (I-m_k e_k^T)^{-1}$, but I am not sure how to expand that result in order to get to $L_k = I + m_ke_k^T$.

Answer 1

The key is that $e_k^T m_k = 0$.

$(I+m_k e_k^T) (I - m_k e_k^T) = I - m_k e_k^T m_k e_k^T = I$.