How to get the unique generalized inverse matrix that we need？

Question

For matrix equation Ax = b (A is a $3×4$ matrix, x is $4×1$ vector , b is $3×1$ vector).

now, we have matrix A , vector b and already know that the third value of x is zero. How can we get the vector x by using the generalized inverse matrix of A .

Answer 1

Let us assume that $G_{4\times 3}$ is a generalized inverse of $A$.

than we know that, $$AGA=A$$ which means that for any $x$ $$AGAx=Ax$$ $$\implies AGb=b\quad(\because Ax=b)$$

which shows that $Gb=x$.

so, once you the generalized inverse of $A$ then $Gb$ will be the solution.

That $x=Gb$ is always a solution.

If you want a unique generalized inverse, Moore-Penrose inverse instead of generalized inverse will do the necessary.