I found the following formulation of a perturbation problem in my research (see attached image). In step 4c to 4d, the equation goes from:
$$|I+A^{-1}e_k\Gamma^T| = 0$$ $$|1+\Gamma^TA^{-1}e_k| = 0$$
Where $|\dot{}|$ is the determinant. Im having a hard time figuring out how that conversion happened.
It also states that: $$\Gamma^TA^{-1}e_k=-1$$, which becomes "a minimization of an undetermined least squares problem over $k$", such that $\hat{k}=argmax_k\{||A^{-1}e_k||\}$. How did they arrive to this conclusion?
Finally, they say:
$$\hat{\Gamma}=\frac{-A^{-1}e_\hat{k}}{e_\hat{k}^TA^{-T}A^{-1}e_\hat{k}}$$
In all of this, $e_k$ is the elementary basis vector for column $k$; $\Gamma$ is an $N\times1$ vector and A is an $N\times N$ matrix.
I am asking this because I am trying to arrive to the solution for the analogous problem:
$$|I+A^{-1}\Gamma e_k^T| = 0$$
