Find a g-inverse of the following matrix such that it does not contain any zero entry.$$\begin{pmatrix} 1 & 2 & 1 \\0 & 1 & 1\\ 1 & 3 & 2 \end{pmatrix}.$$
Work done:
I found a generalized inverse and I try to find the suitable entries to make it nonzero.
for example,
$$G=\begin{bmatrix} 0 & 0 &0\\ 1 & -1 & 0\\ -1 & 2 & 0 \end{bmatrix}$$ Now, How to get a g-inverse which does not have any nonzero entry?
By using the sagemath, I have calculated the g-inverses by varying the entries of $u$ in the expression $G+(I-GA)U$ to get the new g-inverses and after some trial and error work, I found $$G_1=\left(\begin{array}{rrr} 2 & 2 & 3 \\ -1 & -3 & -3 \\ 1 & 4 & 3 \end{array}\right) $$ is a g-inverse with non-zero entries for the the $u=\left(\begin{array}{rrr} 2 & 2 & 3 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right)$.
Is there any other possibility to get the g-inverse with all entries are non-zero in a simple manner?(means without trial and error, because it may take so much time for some matrices)
Definition:
A matrix $G$ is said to be generalized inverse of $A$ if $$AGA=A$$