I would like to ask how free you are when you are calculating generalized inverses? I know, that it is said that there are infinite many of them, but we always usually choose submatrix such that elements are picked in square shape so it is easy to put them back in original matrix.
For example would it really be possible to get next matrix and if not why not and if yes how?: $\begin{bmatrix}1 &0 &0 & 0 & 0\\0 & 0 & 1 & 0 & 0\\0 & 0 & 0 & 0 & 1\\0 & 0 & 0 & 0 & 0\end{bmatrix}$
I'm not sure what version of "generalized inverse" you're talking about. The Moore-Penrose pseudo-inverse of the given matrix is
$$ \left[\matrix{1 & 0 & 0 & 0\cr 0 & 0 & 0 & 0\cr 0 & 1 & 0 & 0\cr 0 & 0 & 0 & 0\cr 0 & 0 & 1 & 0\cr}\right]$$
Given a rank-deficient matrix $A$, here is a method to calculate a generalized inverse $A^-$ such that $$AA^-A = A$$ Eliminating columns/rows $\big($i.e. cols $\#2,4$ and row $\#4\big)$ yields a full-rank reduced matrix $$A_r=\begin{bmatrix}1 &0 &0 \\0 & 1 & 0\\0 & 0 & 1\end{bmatrix}$$ Calculate $A_r^{-1}\,$ then re-insert zero rows/cols $\big($rows $\#2,4$ and col $\#4\big)$ to obtain $$A^- = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \end{bmatrix}$$ This process does not necessarily generate the Moore-Penrose inverse $A^+$ (however for this particular matrix it does). See page 9 of these course notes for some examples.
