I have a question about how the inverse of a gaussian transformation matrix, $M_k = I - m_k e_k^T$, is derived. The derivation I saw in a class is \begin{align} M_k^{-1} =& (I + \bar L)^{-1}\\\\ =& I - \bar L \\\\ =& I - (M_k - I)\\\\ =& 2 I - M_k\\\\ =& I + m_k e_k^T \end{align} where the matrix $\bar L$ contains the entries below the diagonal of $M_k$. How do we know that $(I + \bar L)^{-1} = I - \bar L$? From what I've been able to gather, it is true for atomic matrices in general. But I can't find a proof of it anywhere.
Note: I've seen the proof $(I - m_k e_k^T)(I + m_k e_k^T) = I$ and it makes sense, but it doesn't tell me anything about how $I + m_k e_k^T$ was found in the first place.